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ICE C.7 What proportion of the t-distribution with 19 degrees of freedom falls above -1.79 units?1 EXAMPLE C.8 Find the value of t of freedom where 95% of the distribution lies between -t 18 using the t-table, where t 18 and +t 18. 18 is the cuto๏ฌ for the t-distribution with 18 degrees For a 95% con๏ฌdence interval, we want to ๏ฌnd the cuto๏ฌ t 18 such that 95% of the t-distribution is between -t 18; this is the same as where the two tails have a total area of 0.05. We look in the t-table on page 514, ๏ฌnd the column with area totaling 0.05 in the two tails (third column), and then the row with 18 degrees of freedom: t 18 and t 18 = 2.10. 1We ๏ฌnd the shaded area above -1.79 (we leave the picture to you). The small left tail is between 0.025 and 0.05, so the larger upper region must have an area between 0.95 and 0.975. โ4โ2024โ4โ2024 516 APPENDIX C. DISTRIBUTION TABLES one tail two tails 1 df 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.100 0.200 3.08 1.89 1.64 1.53 1.48 1.44 1.41 1.40 1.38 1.37 1.36 1.36 1.35 1.35 1.34 1.34 1.33 1.33 1.33 1.33 1.32 1.32 1.32 1.32 1.32 1.31 1.31 1.31 1.31 1.31 0.050 0.100 6.31 2.92 2.35 2.13 2.02 1.94 1.89 1.86 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.75 1.74 1.73 1.73 1.72 1.72 1.72 1.71 1.71 1.71 1.71 1.70 1.70 1.70 1.70 0.025 0.050 12.71 4.30
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3.18 2.78 2.57 2.45 2.36 2.31 2.26 2.23 2.20 2.18 2.16 2.14 2.13 2.12 2.11 2.10 2.09 2.09 2.08 2.07 2.07 2.06 2.06 2.06 2.05 2.05 2.05 2.04 0.010 0.020 31.82 6.96 4.54 3.75 3.36 3.14 3.00 2.90 2.82 2.76 2.72 2.68 2.65 2.62 2.60 2.58 2.57 2.55 2.54 2.53 2.52 2.51 2.50 2.49 2.49 2.48 2.47 2.47 2.46 2.46 0.005 0.010 63.66 9.92 5.84 4.60 4.03 3.71 3.50 3.36 3.25 3.17 3.11 3.05 3.01 2.98 2.95 2.92 2.90 2.88 2.86 2.85 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.76 2.76 2.75 โ3โ2โ10123One Tailโ3โ2โ10123One Tailโ3โ2โ10123Two Tails 517 one tail two tails 31 df 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 60 70 80 90 100 150 200 300 400 500 โ 0.100 0.200 1.31 1.31 1.31 1.31 1.31 1.31 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.29 1.29 1.29 1.29 1.29 1.29 1.28 1.28 1.28 1.28 0.050 0.100 1.70 1.69 1.69 1.69 1.69 1.69 1.69 1.69 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.67
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1.67 1.66 1.66 1.66 1.66 1.65 1.65 1.65 1.65 1.65 0.025 0.050 2.04 2.04 2.03 2.03 2.03 2.03 2.03 2.02 2.02 2.02 2.02 2.02 2.02 2.02 2.01 2.01 2.01 2.01 2.01 2.01 2.00 1.99 1.99 1.99 1.98 1.98 1.97 1.97 1.97 1.96 1.96 0.010 0.020 2.45 2.45 2.44 2.44 2.44 2.43 2.43 2.43 2.43 2.42 2.42 2.42 2.42 2.41 2.41 2.41 2.41 2.41 2.40 2.40 2.39 2.38 2.37 2.37 2.36 2.35 2.35 2.34 2.34 2.33 2.33 0.005 0.010 2.74 2.74 2.73 2.73 2.72 2.72 2.72 2.71 2.71 2.70 2.70 2.70 2.70 2.69 2.69 2.69 2.68 2.68 2.68 2.68 2.66 2.65 2.64 2.63 2.63 2.61 2.60 2.59 2.59 2.59 2.58 โ3โ2โ10123One Tailโ3โ2โ10123One Tailโ3โ2โ10123Two Tails 518 APPENDIX C. DISTRIBUTION TABLES C.4 Chi-Square Probability Table A chi-square probability table may be used to ๏ฌnd tail areas of a chi-square distribution. The chi-square table is partially shown in Figure C.5, and the complete table may be found on page 519. When using a chi-square table, we examine a particular row for distributions with di๏ฌerent degrees of freedom, and we identify a range for the area (e.g. 0.025 to 0.05). Note that the chi-square table provides upper tail values, which is di๏ฌerent than the normal and t-distribution tables. Upper tail 2 df 3 4 5 6 7 0
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.3 2.41 3.66 4.88 6.06 7.23 8.38 0.2 3.22 4.64 5.99 7.29 8.56 9.80 0.1 4.61 6.25 7.78 9.24 10.64 12.02 0.05 5.99 7.81 9.49 11.07 12.59 14.07 0.02 7.82 9.84 11.67 13.39 15.03 16.62 0.01 9.21 11.34 13.28 15.09 16.81 18.48 0.005 10.60 12.84 14.86 16.75 18.55 20.28 0.001 13.82 16.27 18.47 20.52 22.46 24.32 Figure C.5: A section of the chi-square table. A complete table is in Appendix C.4. EXAMPLE C.9 Figure C.6(a) shows a chi-square distribution with 3 degrees of freedom and an upper shaded tail starting at 6.25. Use Figure C.5 to estimate the shaded area. This distribution has three degrees of freedom, so only the row with 3 degrees of freedom (df) is relevant. This row has been italicized in the table. Next, we see that the value โ 6.25 โ falls in the column with upper tail area 0.1. That is, the shaded upper tail of Figure C.6(a) has area 0.1. This example was unusual, in that we observed the exact value in the table. In the next examples, we encounter situations where we cannot precisely estimate the tail area and must instead provide a range of values. EXAMPLE C.10 Figure C.6(b) shows the upper tail of a chi-square distribution with 2 degrees of freedom. The area above value 4.3 has been shaded; ๏ฌnd this tail area. The cuto๏ฌ 4.3 falls between the second and third columns in the 2 degrees of freedom row. Because these columns correspond to tail areas of 0.2 and 0.1, we can be certain that the area shaded in Figure C.6(b) is between 0.1 and 0.2. EXAMPLE C.11 Figure C.6(c) shows an upper tail for a chi-square distribution with 5 degrees of freedom and a cuto๏ฌ
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of 5.1. Find the tail area. Looking in the row with 5 df, 5.1 falls below the smallest cuto๏ฌ for this row (6.06). That means we can only say that the area is greater than 0.3. EXAMPLE C.12 Figure C.6(d) shows a cuto๏ฌ of 11.7 on a chi-square distribution with 7 degrees of freedom. Find the area of the upper tail. The value 11.7 falls between 9.80 and 12.02 in the 7 df row. Thus, the area is between 0.1 and 0.2. 519 (a) (c) (b) (d) Figure C.6: (a) Chi-square distribution with 3 degrees of freedom, area above 6.25 shaded. (b) 2 degrees of freedom, area above 4.3 shaded. (c) 5 degrees of freedom, area above 5.1 shaded. (d) 7 degrees of freedom, area above 11.7 shaded. Upper tail 1 df 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 0.3 1.07 2.41 3.66 4.88 6.06 7.23 8.38 9.52 10.66 11.78 12.90 14.01 15.12 16.22 17.32 18.42 19.51 20.60 21.69 22.77 28.17 33.53 44.16 54.72 0.2 1.64 3.22 4.64 5.99 7.29 8.56 9.80 11.03 12.24 13.44 14.63 15.81 16.98 18.15 19.31 20.47 21.61 22.76 23.90 25.04 30.68 36.25 47.27 58.16 0.1 2.71 4.61 6.25 7.78 9.24 10.64 12.02 13.36 14.68 15.99 17.28 18.55 19.81 21.06 22.31 23.54 24.77 25.99 27.20 28.41 34.38 40.26 51.81 63.17 0.05 3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92 18.31 19.68 21.03
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22.36 23.68 25.00 26.30 27.59 28.87 30.14 31.41 37.65 43.77 55.76 67.50 0.02 5.41 7.82 9.84 11.67 13.39 15.03 16.62 18.17 19.68 21.16 22.62 24.05 25.47 26.87 28.26 29.63 31.00 32.35 33.69 35.02 41.57 47.96 60.44 72.61 0.01 6.63 9.21 11.34 13.28 15.09 16.81 18.48 20.09 21.67 23.21 24.72 26.22 27.69 29.14 30.58 32.00 33.41 34.81 36.19 37.57 44.31 50.89 63.69 76.15 0.005 7.88 10.60 0.001 10.83 13.82 12.84 14.86 16.75 18.55 20.28 21.95 23.59 25.19 26.76 28.30 29.82 31.32 32.80 34.27 35.72 37.16 38.58 40.00 46.93 53.67 66.77 79.49 16.27 18.47 20.52 22.46 24.32 26.12 27.88 29.59 31.26 32.91 34.53 36.12 37.70 39.25 40.79 42.31 43.82 45.31 52.62 59.70 73.40 86.66 0510152025051015202505101520250510152025 520 Index 5-number summary, 95 accurate, 259 Addition Rule of disjoint outcomes, 140 alternative hypothesis, 274, 284 anecdotal evidence, 28 ask, 506 associated, 20, 22 average, 27 bar chart segmented, 119 side-by-side, 119 bar chart, 116 Bayesโ Theorem, 167, 164โ167 Bayesian statistics, 167 bias, 255 bimodal, 69 binomial coe๏ฌcient, 201, 212 binomial distribution, 212 binomial formula, 201, 212 bivariate, 59, 69 blind, 47 blocked experiment, 48, 48โ50, 51 blocking, 48, 53 blocks, 48 box plot, 80, 95 side-by-side box plot, 91 case, 16 categorical,
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322 population, 26, 30, 26โ38 population mean, 255 positive association, 21, 60, 69 possum, 504, 508 power, 282, 285 power analysis, 282 practically signi๏ฌcant, 283 precise, 259 prediction, 431 primary, 163 probability, 138, 147, 136โ167, 254 probability distribution, 178 probability of a success, 193, 200 probability of failure 1 โ p, 198 probability of success p, 198 probability sample, see sample probability table, 104 proportion, 27, 30 prospective study, 35 quartile ๏ฌrst quartile (Q1), 80 second quartile (Q2, median), 80 third quartile (Q3), 80 R, 256 R-squared, 451 random assignment, 53 random noise, 126 random numbers, 172 observational study, 30 observational unit, 16 one-sample t-interval, see t-interval for a mean one-sample t-test, see t-test for a mean one-sided, 275 ordinal, 19 psuedo-random numbers, 172 random process, 138, 138โ139 random sampling, 53 random variable, 180, 177โ189 combine, 190 randomization, 127 523 randomized experiment, 30 range, 76, 95 relative frequency, 67, 138, 147, 175 replication, 51 representative, 37 residual, 424, 424โ427, 431 residual plot, 425, 431 response bias, 37 response variable, 29, 423, 431 resume, 507 retrospective studies, 35 right skewed, 69, 95 robust estimates, 87 row totals, 116 rule of complements, 147 run17, 505 run17samp, 505, 507 sample, 26, 30, 26โ38 cluster sampling, 40 convenience sample, 36 multistage cluster sampling, 40 multistage sampling, 40 non-response, 37 non-response bias, 37 random sample, 35โ38 simple random sampling, 38 strata, 40 strati๏ฌed sampling, 40 systematic sampling, 38 sample mean, 255 sample proportion, 194, 270 sample space, 143 sample statistic, 86 sampling distribution, 219, 231, 239, 256 sampling error, 255 sat improve, 507 scatterplot, 20, 59, 69, 431 SD, see standard deviation SE, see standard error second quartile, 80 secondary, 163 sets, 140 shape, 67 shape of the sampling distribution, 307 side-by-side box plot, 91 signi๏ฌcance level
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, 277, 277, 282, 284 signi๏ฌcant, 14 simple random sample, 36, 43 simulated scatter, 508 simulation, 127, 172, 172โ175, 279 single-blind, 47 skew left skewed, 67 right skewed, 67 strongly skewed guideline, 236 symmetric, 67 slope, 451 smallpox, 504 spread, 80, 95 standard deviation, 76, 95, 182, 190 standard deviation of the residuals, 425 standard error, 258, 260, 307 single mean, 233 single proportion, 294 standard normal distribution, 102 standard units, 79 statistic, 27, 30, 26โ30 statistically signi๏ฌcant, 14, 128, 277, 284 stem, 62 stem-and-leaf plot, 62, 69 split stem-and-leaf plot, 62 stem cells, 507 stent30, 503 stent365, 503 stocks 18, 504 strata, 40, 43 strati๏ฌed random sample, 43 stratifying, 53 study participants, 46 success, 193, 200 success-failure condition, 222, 227, 294 suits, 141 summary statistic, 13, 14, 20, 86 symmetric, 67, 69 systematic random sample, 43 t-distribution, 363โ366 t-interval for a di๏ฌerence of means, 399, 397โ402, 409 for a mean, 370, 367โ372, 379 for a mean of di๏ฌerences, 389, 388โ391, 392 for a slope, 467, 477, 465โ477 t-probability table, 514 T-statistic, 375 t-table, 364, 514 t-test for a di๏ฌerence of means, 405, 402โ408, 409 for a mean, 376, 374โ378, 379 for a mean of di๏ฌerences, 386, 384โ388, 392 for a slope, 473, 469โ475, 477 table proportions, 153 tail, 67 test statistic, 104, 284 textbooks, 507, 508 the ๏ฌrst success on the xth trial, 198 third quartile, 80 time series, 464 time series data, 236 transform, 458, 460 transformation, 456, 458, 460 transplant, 505 treatment, 50 treatment group, 12, 14, 46 tree diagram, 163, 163โ167 trial, 193, 200 524 INDEX two-proportion Z-interval, see Z-interval
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for a dif- ference of proportions two-proportion Z-test, see Z-test for a di๏ฌerence of proportions two-sample t-interval, see t-interval for a di๏ฌer- ence of means two-sample t-test, see t-test for a di๏ฌerence of means two-sided, 275 two-way table, 353 Type I Error, 281, 284 Type II Error, 281, 284 ucla textbooks f18, 507, 508 unbiased, 223, 260 unconditional probability, 168 undercoverage bias, 36 unimodal, 69 unit of observation, 16 univariate, 61, 69 variability, 76, 80 variable, 16, 27 variance, 76, 182 Venn diagrams, 141 volunteer sample, 37 volunteers, 46 weighted mean, 74 whiskers, 81 with replacement, 159 without replacement, 158, 168 y-intercept, 451 Z-interval for a di๏ฌerence of proportions, 313, 312โ315, 322 for a proportion, 297, 295โ299, 307 Z-score, 79, 95, 113, 227 Z-test for a di๏ฌerence of proportions, 319, 316โ321, 322 for a proportion, 304, 302โ306, 307 525 Appendix D Technology reference, Formulas, and Inference guide D.1 Technology reference Instructions for the TI-83/84 and the Casio fx-9750GII, and their associated videos. Summarizing 1-variable statistics Entering data Calculating summary statistics. Drawing a box plot Finding normal probabilities Finding area under the normal curve Finding a Z-score that corresponds to a percentile Binomial probabilities Computing the binomial coe๏ฌcient Computing the binomial formula Computing cumulative binomial probabilities Inference for a single proportion 1-proportion Z-interval 1-proportion Z-test Inference for a di๏ฌerence of proportions 2-proportion Z-interval 2-proportion Z-test Chi-square for one-way tables Finding area under chi-square curve Chi-square goodness of ๏ฌt test Chi-square for two-way tables Entering data in a two-way table Chi-square test for homogeneity and independence Finding the expected counts Inference for a single mean 1-sample t-test 1-sample t-interval Inference for a mean of di๏ฌerences 1-sample t-
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test with paired data 1-sample t-interval with paired data Inference for a di๏ฌerence of means 2-sample t-interval 2-sample t-test The least squares regression line Finding the y-intercept, slope, r, and R2 What to do if you get Dim Mismatch Inference for the slope of a regression line page 83 page 84 page 84 page 110 page 112 page 205 page 206 page 206 page 299 page 306 page 315 page 321 page 331 page 337 page 351 page 351 page 351 page 378 page 372 page 388 page 391 page 401 page 407 page 445 page 446 t-interval for the slope t-test for the slope see Graphing Calculator Guides at openintro.org/ahss see Graphing Calculator Guides at openintro.org/ahss D.2 Formulas Descriptive Statistics xi n xi = ยฏx = 1 n sx = 1 n โ 1 (xi โ ยฏx)2 r = 1 n โ 1 xi โ ยฏx yi โ ยฏy sx sy หy = a + bx ยฏy = a + bยฏx b = r sy sx s = (yi โ หyi)2 n โ 2 Inferential Statistics Probability P (A โช B) = P (A) + P (B) โ P (A โฉ B) P (A|B) = P (A โฉ B) P (B) ยตX = E(X) = xi ยท P (xi) ฯX = (xi โ ยตx)2 ยท P (xi) ยตยฏx = ยต ฯยฏx = ฯ โ n If X has a binomial distribution with parameters n and p, then: P (X = x) = n x px(1 โ p)nโx ยตX = np ฯX = np(1 โ p) ยต หp = p ฯ หp = p(1 โ p) n standardized test statistic: point estimate โ null value SE of estimate con๏ฌdence interval: point estimate ยฑ critical value ร SE of estimate parameter point estimate single proportion p หp SE of estimate หp(1โ หp) n when H0: p = p0, use p0(1โp0) n di๏ฌ. of proportions p1 โ p2
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หp1 โ หp2 หp1(1โ หp1) n1 + หp2(1โ หp2) n2 when H0: p1 = p2, use โ หpc (1โ หpc ) 1 n1 + 1 n2 single mean ยต mean of di๏ฌerences ยตdi๏ฌ ยฏx ยฏxdi๏ฌ di๏ฌerence of means ยต1 โ ยต2 ยฏx1 โ ยฏx2 slope of reg. line ฮฒ b sโ n sdi๏ฌโ ndi๏ฌ s2 1 n1 + s2 2 n2 s โ nโ1 sx Chi-square test statistic = (observedโexpected)2 expected D.3 Inference Guide ฮฉ INFERENCE GUIDE CONFIDENCE INTERVALS Use confidence intervals to estimate a parameter with a particular confidence level, C. IDENTIFY: Identify the parameter and the confidence level. CHOOSE: Choose and name the appropriate interval. CHECK: Check that conditions for the procedure are met. CALCULATE: ๐๐: ๐ฉ๐จ๐ข๐ง๐ญ ๐๐ฌ๐ญ๐ข๐ฆ๐๐ญ๐ ยฑ ๐๐ซ๐ข๐ญ๐ข๐๐๐ฅ ๐ฏ๐๐ฅ๐ฎ๐ ร ๐บ๐ฌ ๐จ๐ ๐๐ฌ๐ญ๐ข๐ฆ๐๐ญ๐ ๐๐ = (if applicable) ( ____, ____ ) CONCLUDE: We are C% confident that the true [parameter] is between ____ and ____. (Put the parameter in context.) We have evidence that [โฆ], because [โฆ]. OR We do not have evidence that [โฆ], because [โฆ]. When the parameter is: a single proportion p CHOOSE: 1-Proportion Z-Interval to estimate ๐, or 1-Proportion Z-Test to test ๐ป0: ๐=๐0. CHECK
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: - Data come from a random sample or process. - for CI: ๐๐ฬโฅ10 and ๐(1โ๐ฬ)โฅ10. for Test: ๐๐0โฅ10 and ๐(1โ๐0)โฅ10. CALCULATE: (1-PropZInt or 1-PropZTest) point estimate: sample proportion ๐ฬ SE of estimate: for CI, use โ๐ฬ(1โ๐ฬ)๐ ; for Test, use โ๐0(1โ๐0)๐ When the parameter is: a difference of proportions p1-p2 CHOOSE: 2-Proportion Z-Interval to estimate ๐1โ๐2, or 2-Proportion Z-Test to test ๐ป0: ๐1=๐2. CHECK: - Data come from 2 independent random samples or 2 randomly assigned treatments. - ๐1๐ฬ1โฅ10, ๐1(1โ๐ฬ1)โฅ10, ๐2๐ฬ2โฅ10, ๐2(1โ๐ฬ2)โฅ10. Note: use ๐ฬ๐, the pooled proportion, in place of ๐ฬ1 and ๐ฬ2 when checking condition for the 2-Proportion Z-Test CALCULATE: (2-PropZInt or 2-PropZTest) point estimate: difference of sample proportions ๐ฬ1โ๐ฬ2 SE of estimate: CI, useโ๐ฬ1(1โ๐ฬ1)๐1+๐ฬ2(1โ๐ฬ2)๐2 ; Test, useโ๐ฬ๐(1โ๐๐ฬ) โ1๐1+1๐2 HYPOTHESIS TESTS Use hypothesis tests to test ๐ป0 versus ๐ป๐ด at a particular significance level
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, ฮฑ. IDENTIFY: Identify the hypotheses and the significance level. CHOOSE: Choose and name the appropriate test. CHECK: Check that conditions for the procedure are met. CALCULATE: ๐ฌ๐ญ๐๐ง๐๐๐ซ๐๐ข๐ณ๐๐ ๐ญ๐๐ฌ๐ญ ๐ฌ๐ญ๐๐ญ๐ข๐ฌ๐ญ๐ข๐=๐ฉ๐จ๐ข๐ง๐ญ ๐๐ฌ๐ญ๐ข๐ฆ๐๐ญ๐โ ๐ง๐ฎ๐ฅ๐ฅ ๐ฏ๐๐ฅ๐ฎ๐๐บ๐ฌ ๐จ๐ ๐๐ฌ๐ญ๐ข๐ฆ๐๐ญ๐ ๐๐ = (if applicable) p-value = CONCLUDE: p-value < ฮฑ, so we reject ๐ป0. We have evidence that [๐ป๐ด]. (Put ๐ป๐ด in context.) OR p-value > ฮฑ, so we do NOT reject ๐ป0. We do NOT have evidence that [๐ป๐ด]. (Put ๐ป๐ด in context.) When the parameter is: a single mean ฮผ CHOOSE: 1-Sample T-Interval to estimate ๐, or 1-Sample T-Test to test ๐ป0: ๐=๐0. CHECK: - Data come from a random sample or process. - ๐โฅ30, OR population known to be nearly normal, OR population could to be nearly normal because data has no excessive skew or outliers (draw graph). CALCULATE: (TInterval or T-Test) point estimate: sample mean ๐ฅฬ
SE of estimate: ๐ โ๐ ๐๐=๐โ1 When the parameter is: a difference of means ฮผ1-ฮผ2 CHOOSE: 2-Sample T-Interval
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to estimate ๐1โ๐2, or 2-Sample T-Test to test ๐ป0: ๐1=๐2. CHECK: - Data come from 2 independent random samples or 2 randomly assigned treatments. - ๐1โฅ30 and ๐2โฅ30, OR both populations known to be nearly normal, OR both populations could be nearly normal because both data sets have no excessive skew or outliers (draw 2 graphs). CALCULATE: (2-SampTInt or 2-SampTTest) point estimate: difference of sample means ๐ฅฬ
1โ๐ฅฬ
2 SE of estimate: โ๐ 12๐1+๐ 22๐2 ๐๐: use technology When the parameter is: a mean of differences ฮผdiff CHOOSE: 1-Sample T-Interval to estimate ๐abcc, or 1-Sample T-Test to test ๐ป<: ๐abcc = 0.CHECK: -There is paired data from a random sample or matchedpairs experiment.-๐abcc โฅ 30, OR population of differences known to benearly normal, OR population of differences could benearly normal because observed differences have noexcessive skew or outliers (draw graph of differences).CALCULATE: (TInterval or T-Test) point estimate: mean of sample difference ๐ฅฬ
abcc SE of estimate: _deffVNdeff๐๐=๐abccโ1 When the parameter is: the slope ฮฒ of a regression lineCHOOSE: T-Interval for the slope to estimate ๐ฝ, or T-Test for the slope to test ๐ป<: ๐ฝ = 0.CHECK: -There is (x, y) data from a random sample or experiment.-The residual plot shows no pattern making a linearmodel reasonable. (More specifically, the residualsshould be independent, nearly normal, and haveconstant standard deviation.)CALCULATE: (LinRegTInt or LinRegTTest) point estimate: sample slope ๐ SE of estimate: SE of slope (from computer output) ๐๐=๐โ2 The ฯ2 tests for categorical
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variables: chi-square statistic =โ(๐จ๐๐ฌ๐๐ซ๐ฏ๐๐ M ๐๐ฑ๐ฉ๐๐๐ญ๐๐)๐๐๐ฑ๐ฉ๐๐๐ญ๐๐ When comparing the distribution of one categorical variable to a fixed/specified population distribution CHOOSE: ฯ2 Goodness of Fit Test CHECK: -Data come from a random sample or process.-All expected counts โฅ 5. (To calculate expected counts for each category, multiply the sample size by the expected proportion under ๐ป<.) CALCULATE: (ฯ2GOF-Test) ๐P = ๐๐= # of categories โ 1 When comparing the distribution of a categorical variable across 2 or more populations/treatments CHOOSE: ฯ2 Test for Homogeneity CHECK: -Data come from 2 or more independent random samples or 2 or more randomly assigned treatments.-All expected counts โฅ 5. (Calculate expected counts and verify this to be true.) CALCULATE: (ฯ2-Test, then 2ND MATRIX,EDIT,2:[B] to find expected counts) ๐P = ๐๐ = (# of rows โ 1)(# of cols โ 1) When looking for association or dependence between two categorical variables CHOOSE: ฯ2 Test for Independence CHECK: -Data come from a random sample or process.-All expected counts โฅ 5. (Calculate expected counts and verify this to be true.) CALCULATE: (ฯ2-Test, then 2ND MATRIX,EDIT,2:[B] to find expected counts) ๐P = ๐๐ = (# of rows โ 1)(# of cols โ 1) ______________________________________________________________________________________________________________ โ3, โ2, โ1, The set of rational numbers is written as โง positive integers zero 0, 1, 2, 3, โฏ โจm โฌ. Notice from the definition that rational n |m and n are integers and n โ 0 numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and
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the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1. โญ โซ โฉ Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either: 1. a terminating decimal: 15 8 = 1.875, or This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 11 2. a repeating decimal: 4 11 = 0.36363636 โฆ = 0.36 We use a line drawn over the repeating block of numbers instead of writing the group multiple times. Example 1.1 Writing Integers as Rational Numbers Write each of the following as a rational number. a. 7 b. 0 c. โ8 Solution Write a fraction with the integer in the numerator and 1 in the denominator. a. b. โ8 = โ 8 1 1.1 Write each of the following as a rational number. a. 11 b. 3 c. โ4 Example 1.2 Identifying Rational Numbers Write each of the following rational numbers as either a terminating or repeating decimal. a. โ 5 7 b. c. 15 5 13 25 Solution Write each fraction as a decimal by dividing the numerator by the denominator. a. โ 5 7 โโโ = โ0.714285, a repeating decimal 12 Chapter 1 Prerequisites b. c. 15 5 13 25 = 3 (or 3.0), a terminating decimal = 0.52, a terminating decimal 1.2 Write each of the following rational numbers as either a terminating or repeating decimal. a. b. 68 17 8 13 c. โ 17 20 Irrational Numbers At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even 3, but was something else. Or a garment 2 maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is
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not rational. So we write this as shown. {h|h is not a rational number} Example 1.3 Differentiating Rational and Irrational Numbers Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal. a. b. c. d. e. 25 33 9 11 17 34 0.3033033303333 โฆ Solution a. b. 25 : This can be simplified as 25 = 5. Therefore, 25 is rational. 33 9 : Because it is a fraction, 33 9 is a rational number. Next, simplify and divide. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 13 33 9 11 = 33 9 3 ยฏ = 3. 6 = 11 3 So, 33 9 is rational and a repeating decimal. c. d. 11 : This cannot be simplified any further. Therefore, 11 is an irrational number. 17 34 : Because it is a fraction, 17 34 is a rational number. Simplify and divide. 17 34 1 = 17 34 2 = 1 2 = 0.5 So, 17 34 is rational and a terminating decimal. e. 0.3033033303333 โฆ is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number. Determine whether each of the following numbers is rational or irrational. If it is rational, determine 1.3 whether it is a terminating or repeating decimal. a. b. c. d. e. 7 77 81 4.27027002700027 โฆ 91 13 39 Real Numbers Given any number n, we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or โ). Zero is considered neither positive nor negative. The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other
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basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in Figure 1.2. Figure 1.2 The real number line Example 1.4 14 Chapter 1 Prerequisites Classifying Real Numbers Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line? a. โ 10 3 b. 5 c. โ 289 d. โ6ฯ e. 0.615384615384 โฆ Solution a. โ 10 3 is negative and rational. It lies to the left of 0 on the number line. b. 5 is positive and irrational. It lies to the right of 0. c. โ 289 = โ 172 = โ17 is negative and rational. It lies to the left of 0. d. โ6ฯ is negative and irrational. It lies to the left of 0. e. 0.615384615384 โฆ is a repeating decimal so it is rational and positive. It lies to the right of 0. Classify each number as either positive or negative and as either rational or irrational. Does the number lie 1.4 to the left or the right of 0 on the number line? a. 73 b. โ11.411411411 โฆ c. 47 19 d. โ 5 2 e. 6.210735 Sets of Numbers as Subsets Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 1.3. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 15 Figure 1.3 Sets of numbers N: the set of natural numbers W: the set of whole numbers I: the set of integers Q: the set of rational numbers Qยด: the set of irrational numbers Sets of Numbers The set of natural numbers includes the numbers used for counting: {1, 2, 3,...}. The set of whole numbers is the set of natural numbers plus zero: {0, 1, 2, 3,...}. The set of integers adds the negative
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natural numbers to the set of whole numbers: {..., โ3, โ2, โ1, 0, 1, 2, 3,...}. The set of rational numbers includes fractions written as โง โจm n |m and n are integers and n โ 0 โฉ โซ โฌ. โญ The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: {h|h is not a rational number}. Example 1.5 Differentiating the Sets of Numbers Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Qโฒ). a. b. c. 36 8 3 73 d. โ6 e. 3.2121121112 โฆ Solution 16 Chapter 1 Prerequisites a. 36 = 6 ยฏ = 2. 6 b. 8 3 c. 73 d. โ6 e. 3.2121121112... โฒ X X Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), 1.5 and/or irrational number (Qโฒ). a. โ 35 7 b. 0 c. d. e. 169 24 4.763763763 โฆ Performing Calculations Using the Order of Operations When we multiply a number by itself, we square it or raise it to a power of 2. For example, 42 = 4 โ
4 = 16. We can raise any number to any power. In general, the exponential notation an means that the number or variable a is used as a factor n times. an = a โ
a โ
a โ
โฆ โ
a In this notation, an is read as the nth power of a, where a is called the base and n is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, 24 + 6 โ
2 3 โ 42 is a mathematical expression. n factors To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions. Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated
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as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 17 The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right. Letโs take a look at the expression provided. There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify 42 as 16. 24 + 6 โ
2 3 โ 42 Next, perform multiplication or division, left to right. Lastly, perform addition or subtraction, left to right. 24 + 6 โ
2 3 24 + 6 โ
2 3 โ 42 โ 16 24 + 6 โ
2 3 โ 16 24 + 4 โ 16 24 + 4 โ 16 28 โ 16 12 Therefore, 24 + 6 โ
2 3 โ 42 = 12. For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result. Order of Operations Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS: P(arentheses) E(xponents) M(ultiplication) and D(ivision) A(ddition) and S(ubtraction) Given a mathematical expression, simplify it using the order of operations. 1. Simplify any expressions within grouping symbols. 2. Simplify any expressions containing exponents or radicals. 3. Perform any multiplication and division in order, from left to right. 4. Perform any addition and subtraction in order, from left to right. Example 1.6 Using the Order of Operations Use the order of operations to evaluate each of the following expressions. 18 Chapter 1 Prerequisites a. b. c. d. e. (3 โ
2)2 โ 4(6 + 2) 52 โ 4 7 โ 11 โ 2 6 โ |5 โ 8| + 3(4 โ 1) 14 โ 3 โ
2 2 โ
5 โ 32 โก ๏ฟฝ
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๏ฟฝ๏ฟฝ(6 โ 3) โ 42โค 7(5 โ
3) โ 2 โฆ + 1 Solution (3 โ
2)2 โ 4(6 + 2) = (6)2 โ 4(8) = 36 โ 4(8) = 36 โ 32 = 4 Simplify parentheses Simplify exponent Simplify multiplication Simplify subtraction 52 7 โ 11 โ 2 = 52 โ 4 7 = 52 โ 4 7 = 25 โ 4 7 โ 3 = 21 7 = 3 โ 3 = 0 โ 9 Simplify grouping symbols (radical) โ 3 โ 3 Simplify radical Simplify exponent Simplify subtraction in numerator Simplify division Simplify subtraction Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped. 6 โ |5 โ 8| + 3(4 โ 1) = 6 โ |โ3| + 3(3) = 6 โ 3 + 3(3 = 12 Simplify inside grouping symbols Simplify absolute value Simplify multiplication Simplify subtraction Simplify addition a. b. c. d. 14 โ 3 โ
2 2 โ
5 โ 32 = 14 โ 3 โ
2 2 โ
5 โ 9 = 14 โ 6 10 โ 9 = 8 1 = 8 Simplify exponent Simplify products Simplify diffe ences Simplify quotient In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 19 e. โก โฃ(6 โ 3) โ 42โค 7(5 โ
3) โ 2 โก โฃ(3) โ 42โค โฆ + 1 = 7(15) โ 2 โฆ + 1 Simplify inside parentheses = 7(15) โ 2(3 โ 16) + 1 = 7(15) โ 2(โ13) + 1 = 105 + 26 + 1 = 132 Simplify exponent Subtract Multiply Add 1.6 Use the order of operations to evaluate each of the following expressions. a. 52 โ 42 + 7(5 โ 4)2 b. d. e. |1.8 โ 4.3
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| + 0.4 15 + 10 โก โฃ5 โ
32 โ 72โค 1 โ
92 2 โฆ + 1 3 โค โก โฃ(3 โ 8)2 โ 4 โฆ โ (3 โ 8) Using Properties of Real Numbers For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics. Commutative Properties The commutative property of addition states that numbers may be added in any order without affecting the sum. We can better see this relationship when using real numbers. a + b = b + a (โ2) + 7 = 5 and 7 + (โ2) = 5 Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product. Again, consider an example with real numbers. a โ
b = b โ
a (โ11) โ
(โ4) = 44 and (โ4) โ
(โ11) = 44 It is important to note that neither subtraction nor division is commutative. For example, 17 โ 5 is not the same as 5 โ 17. Similarly, 20 รท 5 โ 5 รท 20. Associative Properties The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same. 20 Chapter 1 Prerequisites Consider this example. a(bc) = (ab)c (3 โ
4) โ
5 = 60 and 3 โ
(4 โ
5) = 60 The associative property of addition tells us that numbers may be grouped differently without affecting the sum. This property can be especially helpful when dealing with negative integers. Consider this example. a + (b + c) = (a + b) + c โก โฃ15 + (โ9)โค โฆ + 23 = 29 and 15 + โก โฃ(โ9) + 23โค โฆ = 29 Are subtraction and division associative? Review these examples. 8 โ (3 โ 15) =? (8
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โ 3) โ 15 8 โ ( โ 12) = 5 โ 15 64 รท (8 รท 4) =? 64 รท 2 =? (64 รท 8) รท 4 8 รท 4 20 โ 20 โ 10 32 โ 2 As we can see, neither subtraction nor division is associative. Distributive Property The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum. a โ
(b + c) = a โ
b + a โ
c This property combines both addition and multiplication (and is the only property to do so). Let us consider an example. Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by โ7, and adding the products. To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example. 6 + (3 โ
5) =? 6 + (15) =? (6 + 3) โ
(6 + 5) (9) โ
(11) 21 โ 99 Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction. A special case of the distributive property occurs when a sum of terms is subtracted. a โ b = a + (โb) For example, consider the difference 12 โ (5 + 3). We can rewrite the difference of the two terms 12 and (5 + 3) by turning the subtraction expression into addition of the opposite. So instead of subtracting (5 + 3), we add the opposite. Now, distribute โ1 and simplify the result. 12 + (โ1) โ
(5 + 3) 12 โ (5 + 3) = 12 + (โ1) โ
(5 + 3) = 12 + [(โ1) โ
5 + (โ1) โ
3] = 12 + (โ8) = 4 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 21 This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last
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example. 12 โ (5 + 3) = 12 + (โ5 โ 3) = 12 + (โ8) = 4 Identity Properties The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number. a + 0 = a The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number. For example, we have (โ6) + 0 = โ6 and 23 โ
1 = 23. There are no exceptions for these properties; they work for every real number, including 0 and 1. a โ
1 = a Inverse Properties The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denotedโa, that, when added to the original number, results in the additive identity, 0. For example, if a = โ8, the additive inverse is 8, since (โ8) + 8 = 0. a + (โa) = 0 The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted 1 a, that, when multiplied by the original number, results in the multiplicative identity, 1. a โ
1 a = 1 For example, if a = โ 2 3, the reciprocal, denoted 1 a, is โ 3 2 because a โ
1 a = โ โโ 2 3 โ โ โ
โ โโ 3 2 โ โ = 1 Properties of Real Numbers The following properties hold for real numbers a, b, and c. 22 Chapter 1 Prerequisites Commutative Property Associative Property Distributive Property Identity Property Inverse Property Addition Multiplication b + c) = (a + b) + c a(bc) = (ab)c a โ
(b + c) = a โ
b + a โ
c There exists a unique real number called the additive identity, 0, such that, for any real number a There exists a unique real number called the multiplicative identity, 1, such that, for any real number Every real number a has an additive
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inverse, or opposite, denoted โa, such that a + (โa) = 0 Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted 1 a, such that โ 1 โ = 1 a a โ
โ โ Example 1.7 Using Properties of Real Numbers Use the properties of real numbers to rewrite and simplify each expression. State which properties apply. a. b. c. d. e. 3 โ
6 + 3 โ
4 (5 + 8) + (โ8) 6 โ (15 + 9 100 โ
โก โฃ0.75 + (โ2.38)โค โฆ Solution a6 + 4) = 3 โ
10 = 30 Distributive property Simplify Simplify This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 23 b. c. d. e. (5 + 8) + (โ8) = 5 + [8 + (โ8)] = 5 + 0 = 5 Associative property of addition Inverse property of addition Identity property of addition 6 โ (15 + 9) = 6 + [(โ15) + (โ9)] = 6 + (โ24) = โ18 Distributive property Simplify Simplify Commutative property of multiplication Associative property of multiplication Inverse property of multiplication Identity property of multiplication 100 โ
[0.75 + ( โ 2.38)] = 100 โ
0.75 + 100 โ
(โ2.38) = 75 + (โ238) = โ163 Distributive property Simplify Simplify 1.7 Use the properties of real numbers to rewrite and simplify each expression. State which properties apply. a. b. c. d. e. โ โโ 23 5 โ โ โ
โก โฃ11 โ
โ โโ 5 23 โค โ โฆ โ 5 โ
(6.2 + 0.4) 18 โ (7โ15) 17 18 + โ
โก โฃ 4 9 + โ โโ 17 18 โค โ โฆ โ 6 โ
(โ3) + 6 โ
3 Evaluating Algebraic Expressions So far, the mathematical expressions we have
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seen have involved real numbers only. In mathematics, we may see expressions such as x + 5, 4 ฯr 3, or 2m3 n2. In the expression x + 5, 5 is called a constant because it does not vary 3 and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way. (โ3)5 = (โ3) โ
(โ3) โ
(โ3) โ
(โ3) โ
(โ3) (2 โ
7)3 = (2 โ
7) โ
(2 โ
7) โ
(2 โ
7) x5 = yz)3 = (yz) โ
(yz) โ
(yz) In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables. 24 Chapter 1 Prerequisites Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before. Example 1.8 Describing Algebraic Expressions List the constants and variables for each algebraic expression. a. x + 5 b. c. ฯr 3 4 3 2m3 n2 Solution Constants Variables a. x + 5 5 ฯr 3 b. 2m3 n2 2 m, n 1.8 List the constants and variables for each algebraic expression. a. 2ฯr(r + h) b. 2(L + W) c. 4y3 + y Example 1.9 Evaluating an Algebraic Expression at Different Values Evaluate the expression 2x โ 7 for each value for x. a. b. c This content is available
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for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 25 d. x = โ4 Solution a. Substitute 0 for x. b. Substitute 1 for x. c. Substitute 1 2 for x. d. Substitute โ4 for x. 2x โ 7 = 2(0) โ 7 = 0 โ 7 = โ7 2x โ 7 = 2(1) โ 7 = 2 โ 7 = โ5 2x โ 6 2x โ 7 = 2( โ 4) โ 7 = โ8 โ 7 = โ15 1.9 Evaluate the expression 11 โ 3y for each value for y. a. b. c. d5 Example 1.10 Evaluating Algebraic Expressions Evaluate each expression for the given values. a. b. c. d. e. x + 5 for x = โ5 t 2tโ1 for t = 10 ฯr 3 for r = 5 4 3 a + ab + b for a = 11, b = โ8 2m3 n2 for m = 2, n = 3 26 Chapter 1 Prerequisites Solution a. Substitute โ5 for x. b. Substitute 10 for t. x + 5 = (โ5) + 5 = 0 t 2t โ 1 = = (10) 2(10) โ 1 10 20 โ 1 = 10 19 c. Substitute 5 for r. 4 3 d. Substitute 11 for a and โ8 for b. ฯ(5)3 ฯr 3 = 4 3 = 4 ฯ(125) 3 = 500 3 ฯ e. Substitute 2 for m and 3 for n. a + ab + b = (11) + (11)(โ8) + (โ8) = 11 โ 88 โ 8 = โ85 2m3 n2 = 2(2)3 (3)2 = 2(8)(9) = 144 = 12 1.10 Evaluate each expression for the given values. y + 3 y โ 3 for y = 5 7 โ 2t for t = โ2 ฯr 2 for r = 11 1 3 3 โp2 qโ โ โ for p = โ2, q = 3 4(m โ n) โ 5(n โ m) for. b. c. d. e. Formulas An equation is a mathematical statement indicating that two expressions are equal.
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The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation 2x + 1 = 7 has the unique solution x = 3 because when we substitute 3 for x in the equation, we obtain the true statement 2(3) + 1 = 7. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 27 A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area A of a circle in terms of the radius r of the circle: A = ฯr 2. For any value of r, the area A can be found by evaluating the expression ฯr 2. Example 1.11 Using a Formula A right circular cylinder with radius r and height h has the surface area S (in square units) given by the formula S = 2ฯr(r + h). See Figure 1.4. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of ฯ. Figure 1.4 Right circular cylinder Solution Evaluate the expression 2ฯr(r + h) for r = 6 and h = 9. S = 2ฯr(r + h) = 2ฯ(6)[(6) + (9)] = 2ฯ(6)(15) = 180ฯ The surface area is 180ฯ square inches. 28 Chapter 1 Prerequisites 1.11 A photograph with length L and width W is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm2) is found to be A = (L + 16)(W + 16) โ L โ
W. See Figure 1.5. Find the area of a matte for a photograph with length 32 cm and width 24 cm. Figure 1.5 Simplifying Algebraic Expressions Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions. Example 1.
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12 Simplifying Algebraic Expressions Simplify each algebraic expression. a. b. c. d. 3x โ 2y + x โ 3y โ 7 2r โ 5(3 โ r) + 4 โ โ4t โ 5 4 sโ โ โ โ โ 2 3 t + 2sโ โ 2mn โ 5m + 3mn + n Solution a. b. 3x โ 2y + x โ 3y โ 7 = 3x + x โ 2y โ 3y โ 7 = 4x โ 5y โ 7 Commutative property of addition Simplify 2r โ 5(3 โ r) + 4 = 2r โ 15 + 5r + 4 = 2r + 5y โ 15 + 4 = 7r โ 11 Distributive property Commutative property of addition Simplify This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 29 โ โt โ 5 4t โ 4 4 sโ โ โ โ โ 2 3 c. d. t + 2sโ t โ 2s โ = 4t โ 5 4 = 4t โ 2 3 t โ 13 4 = 10 3 s โ 2 3 t โ 5 s โ 2s 4 s Distributive property Commutative property of addition Simplify mn โ 5m + 3mn + n = 2mn + 3mn โ 5m + n = 5mn โ 5m + n Commutative property of addition Simplify 1.12 Simplify each algebraic expression. a. b. c 4pโ โq โ 1โ โ + qโ โ1 โ pโ โ d. 9r โ (s + 2r) + (6 โ s) Example 1.13 Simplifying a Formula A rectangle with length L and width W has a perimeter P given by P = L + W + L + W. Simplify this expression. Solution = 2L + 2W P = 2(L + W) Commutative property of addition Simplify Distributive property 1.13 If the amount P is deposited into an account paying simple interest r for time t, the total value of the deposit A is given by A = P + Prt. Simplify the
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expression. (This formula will be explored in more detail later in the course.) Access these online resources for additional instruction and practice with real numbers. โข Simplify an Expression (http://openstaxcollege.org/l/simexpress) โข Evaluate an Expression1 (http://openstaxcollege.org/l/ordofoper1) โข Evaluate an Expression2 (http://openstaxcollege.org/l/ordofoper2) 30 Chapter 1 Prerequisites 1.1 EXERCISES Verbal 1. Is 2 an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category. 2. What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for? What do the Associative Properties allow us to do when 3. following the order of operations? Explain your answer. Numeric 22. 23. (12 รท 3 ร 3)2 25 รท 52 โ 7 24. (15 โ 7) ร (3 โ 7) 25. 2 ร 4 โ 9(โ1) 26. 42 โ 25 ร 1 5 27. 12(3 โ 1) รท 6 For the following exercises, simplify the given expression. Algebraic 4. 10 + 2 ร (5 โ 3) 5. 6. 7. 8. 9. 10. 11. 12. 6 รท 2 โ โ โ81 รท 32โ โ 18 + (6 โ 8)3 โ2 ร โก โฃ16 รท (8 โ 4)2(5 โ 8) 4 + 6 โ 10 รท 2 12 รท (36 รท 9) + 6 (4 + 5)2 รท 3 13. 3 โ 12 ร 2 + 19 14. 2 + 8 ร 7 รท 4 15. 16. 5 + (6 + 4) โ 11 9 โ 18 รท 32 17. 14 ร 3 รท 7 โ 6 18. 9 โ (3 + 11) ร 2 19. 6 + 2 ร 2 โ 1 20. 64 รท (8 + 4 ร 2) 21. โ22โ โ 9 + 4 โ This content is available for free at https://cnx.org/content/col11758/1.5 For the following exercises, solve for the variable. 28. 8(x + 3) =
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64 29. 4y + 8 = 2y 30. (11a + 3) โ 18a = โ4 31. 4z โ 2z(1 + 4) = 36 32. 33. 4y(7 โ 2)2 = โ200 โ(2x)2 + 1 = โ3 34. 8(2 + 4) โ 15b = b 35. 2(11c โ 4) = 36 36. 4(3 โ 1)x = 4 37. 1 4 โ โ8w โ 42โ โ = 0 For the following exercises, simplify the expression. 38. 4x + x(13 โ 7) 39. 40. 2y โ (4)2 y โ 11 a 23(64) โ 12a รท 6 41. 8b โ 4b(3) + 1 42. 5l รท 3l ร (9 โ 6) 7z โ 3 + z ร 62 43. 44. Chapter 1 Prerequisites 4 ร 3 + 18x รท 9 โ 12 45. โy + 8โ 9โ โ โ 27 46. โ t โ 4 โ 2 โ โ 9 6 47. 6 + 12b โ 3 ร 6b 48. 49. 18y โ 2โ โ1 + 7yโ โ 2 โ โ โ โ 4 9 ร 27x 50. 8(3 โ m) + 1(โ8) 51. 9x + 4x(2 + 3) โ 4(2x + 3x) 52. 52 โ 4(3x) For the following exercise, solve the given problem. 31 Ramon runs the marketing department at his company. 59. His department gets a budget every year, and every year, he must spend the entire budget without going over. If he spends less than the budget, then his department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. He must that 2,500,000 โ x = 0. What property of addition tells us what the value of x must be? budget spend such the Technology For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth. 60. 61. 0.5(12.3)2 โ
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48x = 3 5 (0.25 โ 0.75)2 x โ 7.2 = 9.9 Real-World Applications Extensions If a whole number is not a natural number, what must 62. the number be? Determine whether the statement is true or false: The 63. multiplicative inverse of a rational number is also rational. Determine whether the statement is true or false: The 64. product of a rational and irrational number is always irrational. Determine whether 65. rational or irrational: โ18 โ 4(5)(โ1). the simplified expression is Determine whether 66. rational or irrational: โ16 + 4(5) + 5. the simplified expression is The division of two whole numbers will always result 67. in what type of number? What property of real numbers would simplify the 68. following expression: 4 + 7(x โ 1)? For the following exercises, consider this scenario: Fred earns $40 mowing lawns. He spends $10 on mp3s, puts half of what is left in a savings account, and gets another $5 for washing his neighborโs car. Write the expression that represents the number of in his savings 53. dollars Fred keeps (and does not put account). Remember the order of operations. 54. How much money does Fred keep? For the following exercises, solve the given problem. According to the U.S. Mint, the diameter of a quarter is 55. 0.955 inches. The circumference of the quarter would be the diameter multiplied by ฯ. Is the circumference of a quarter a whole number, a rational number, or an irrational number? Jessica and her roommate, Adriana, have decided to 56. share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact? For the following exercises, consider this scenario: There is a mound of g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel. 57. Write the equation that describes the situation. 58. Solve for g. 32 Chapter 1 Prerequisites 1.2 | Exponents and Scientific Notation Learning Objectives
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In this section students will: 1.2.1 Use the product rule of exponents. 1.2.2 Use the quotient rule of exponents. 1.2.3 Use the power rule of exponents. 1.2.4 Use the zero exponent rule of exponents. 1.2.5 Use the negative rule of exponents. 1.2.6 Find the power of a product and a quotient. 1.2.7 Simplify exponential expressions. 1.2.8 Use scientific notation. Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number. Using a calculator, we enter 2,048 ร 1,536 ร 48 ร 24 ร 3,600 and press ENTER. The calculator displays 1.304596316E13. What does this mean? The โE13โ portion of the result represents the exponent 13 of ten, so there are a maximum of approximately 1.3 ร 1013 bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers. Using the Product Rule of Exponents Consider the product x3 โ
x4. Both terms have the same base, x, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression. 3 factors x3 โ
x4 = x โ
x โ
x 4 factors โ
= x7 7 factors The result is that x3 โ
x4 = x3 + 4 = x7. Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents. Now consider
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an example with real numbers. am โ
an = am + n 23 โ
24 = 23 + 4 = 27 We can always check that this is true by simplifying each exponential expression. We find that 23 is 8, 24 is 16, and 27 is 128. The product 8 โ
16 equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents. The Product Rule of Exponents For any real number a and natural numbers m and n, the product rule of exponents states that This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites Example 1.14 Using the Product Rule am โ
an = am + n 33 (1.1) Write each of the following products with a single base. Do not simplify further. a. b. c. t 5 โ
t 3 (โ3)5 โ
(โ3) x2 โ
x5 โ
x3 Solution Use the product rule to simplify each expression. a. b. c (โ3)5 โ
(โ3) = (โ3)5 โ
(โ3)1 = (โ3)5 + 1 = (โ3)6 x2 โ
x5 โ
x3 At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two. x2 โ
x5 โ
x3 = โx2 โ
x5โ โ โ โ
x3 = โ โx2 + 5โ โ โ
x3 = x7 โ
x3 = x7 + 3 = x10 Notice we get the same result by adding the three exponents in one step. x2 โ
x5 โ
x3 = x2 + 5 + 3 = x10 1.14 Write each of the following products with a single base. Do not simplify further. a. b. c Using the Quotient Rule of Exponents The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression
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such as ym yn, where m > n. Consider the example y9 y5. Perform the division by canceling common factors. 34 Chapter 1 Prerequisites y9 y5 = = = = y4 Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend. am an = am โ n In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents. y9 y5 = y9 โ 5 = y4 For the time being, we must be aware of the condition m > n. Otherwise, the difference m โ n could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers. The Quotient Rule of Exponents For any real number a and natural numbers m and n, such that m > n, the quotient rule of exponents states that am an = am โ n (1.2) Example 1.15 Using the Quotient Rule Write each of the following products with a single base. Do not simplify further. a. b. c. (โ2)14 (โ2)9 t 23 t 15 5 โ โz 2โ โ z 2 Solution Use the quotient rule to simplify each expression. a. b. (โ2)14 (โ2)9 = (โ2)14 โ 9 = (โ2)5 t 23 t 15 = t 23 โ 15 = t 8 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 35 c. 5 โ โz 2โ โ z 2 = โ โz 2โ โ 5 โ 1 = โ โz 2โ โ 4 1.15 Write each of the following products with a single base. Do not simplify further. a. b. c. s75 s68 (โ3)6 โ3 5 3 โe f 2โ โ โ โe f 2โ โ โ Using the Power Rule of Exponents Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use
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the power rule of exponents. Consider the expression โ โx2โ โ 3. The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3. 3 โ โx2โ โ = 3 factors โ โx2โ โ โx2โ โ โ
โ โ
โ โx2โ โ 3 factors โ โงโฉโจ2 factorsโ โ x โ
x โ โ
โ โ โ โงโฉโจ2 factorsโ โ x โ
x โ โ โ = โงโฉโจ2 factors = x6 The exponent of the answer is the product of the exponents: โ โx2โ โ 3 = x2 โ
3 = x6. In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents. (am ) = am โ
n n Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents. Product Rule 53 โ
54 = 53 + 4 x5 โ
x2 = x5 + 2 (3a)7 โ
(3a)10 = (3a)7 + 10 = 57 = x7 = (3a)17 but but but (53)4 (x5)2 ((3a)7)10 Power Rule = = = = 512 = x10 53 โ
4 x5 โ
2 (3a)7 โ
10 = (3a)70 The Power Rule of Exponents For any real number a and positive integers m and n, (am ) n the power rule of exponents states that = am โ
n (1.3
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) 36 Chapter 1 Prerequisites Example 1.16 Using the Power Rule Write each of the following products with a single base. Do not simplify further. a. b. c. 7 โx2โ โ โ 3 โ(2t)5โ โ โ 11 โ(โ3)5โ โ โ Solution Use the power rule to simplify each expression. a. b. c. 7 โx2โ โ โ = x2 โ
7 = x14 3 โ(2t)5โ โ โ = (2t)5 โ
3 = (2t)15 11 โ โ(โ3)5โ โ = (โ3)5 โ
11 = (โ3)55 1.16 Write each of the following products with a single base. Do not simplify further. a. b. c. 3 โ โ โ3yโ โ โ 8โ โ 7 โt 5โ โ โ 4 โ โ(โg)4โ โ Using the Zero Exponent Rule of Exponents Return to the quotient rule. We made the condition that m > n so that the difference m โ n would never be zero or negative. What would happen if m = n? In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. To see how this is done, let us begin with an example If we were to simplify the original expression using the quotient rule, we would have If we equate the two answers, the result is t 0 = 1. This is true for any nonzero real number, or any variable representing a real number. a0 = 1 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 37 The sole exception is the expression 00. This appears later in more advanced courses, but for now, we will consider the value to be undefined. The Zero Exponent Rule of Exponents For any nonzero real number a, the zero exponent rule of exponents states that a0 = 1 (1.4) Example 1.17 Using the
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Zero Exponent Rule Simplify each expression using the zero exponent rule of exponents. a. c3 c3 b. โ3x5 x5 c. d. 4 โ โj2 kโ โ โ โj2 kโ โ โ
โ โj2 kโ โ 3 2 5 โ โrs2โ โ 2 โ โrs2โ โ Solution Use the zero exponent and other rules to simplify each expression. a. b. c3 c3 = c3 โ 3 = c3 โ 3 = c3 โ 3 โ3x5 x5 = โ3 โ
x5 x5 = โ3 โ
x5 โ 5 = โ3 โ
x0 = โ3 โ
1 = โ3 38 Chapter 1 Prerequisites c. d. 4 โj2 kโ โ โ โ โj2 kโ โ โ
โj2 kโ โ โ 3 = 4 โj2 kโ โ โ 1 + 3 โj2 kโ โ โ Use the product rule in the denominator. 4 4 โj2 kโ โ โ โj2 kโ โ โ 4 โ 4 โj2 kโ โ โ 0 โj2 kโ โ โ = = = = 1 Simplify. Use the quotient rule. Simplify. 2 โ 2 โrs2โ โ โ Use the quotient rule. 2 5 โrs2โ โ โ 2 = 5 โ โrs2โ โ 0 = 5 โ โrs2โ โ = 5 โ
1 = 5 Simplify. Use the zero exponent rule. Simplify. 1.17 Simplify each expression using the zero exponent rule of exponents. a. b. t 7 t 7 11 11 โ โde2๏ฟฝ
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๏ฟฝ โ โ โde2โ โ 2 c. w4 โ
w2 w6 d Using the Negative Rule of Exponents Another useful result occurs if we relax the condition that m > n in the quotient rule even further. For example, can we simplify h3 h5? When m < n โthat is, where the difference m โ n is negativeโwe can use the negative rule of exponents to simplify the expression to its reciprocal. Divide one exponential expression by another with a larger exponent. Use our example, h3 h5. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 39 h3 h5 = = = h2 If we were to simplify the original expression using the quotient rule, we would have h3 h5 = h3 โ 5 hโ2 = Putting the answers together, we have hโ2 = 1 h2. This is true for any nonzero real number, or any variable representing a nonzero real number. A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction barโfrom numerator to denominator or vice versa. an and an We have shown that the exponential expression an is defined when n is a natural number, 0, or the negative of a natural number. That means that an is defined for any integer n. Also, the product and quotient rules and all of the rules we will look at soon hold for any integer n. = 1 aโn = 1 aโn The Negative Rule of Exponents For any nonzero real number a and natural number n, the negative rule of exponents states that aโn = 1 an (1.5) Example 1.18 Using the Negative Exponent Rule Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents. a. b. c. ฮธ 3 ฮธ 10 z2 โ
z z4 4 8 โโ5t 3โ โ โ โโ5t 3โ โ โ Solution 40 Chapter 1 Prerequisites a. b. c. ฮธ 3 ฮธ 10 = ฮธ 3 โ 10 = ฮธ โ7 = 1 ฮธ 7 z2 + 1 z2 โ
z z4 = z4 =
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z3 z4 = z3 โ 4 = zโ1 = 1 z 4 โโ5t 3โ โ โ 8 = โโ5t 3โ โ โ โโ5t 3โ โ โ 4 โ 8 โโ5t 3โ โ โ = โ4 = 1 โโ5t 3โ โ โ 4 Write each of the following quotients with a single base. Do not simplify further. Write answers with 1.18 positive exponents. a. b. c. (โ3t)2 (โ3t)8 f 47 f 49 โ
f 2k 4 5k 7 Example 1.19 Using the Product and Quotient Rules Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents. a. b. c. b2 โ
bโ8 (โx)5 โ
(โx)โ5 โ7z (โ7z)5 Solution a. b. c. b2 โ
bโ8 = b2 โ 8 = bโ6 = 1 b6 (โx)5 โ
(โx)โ5 = (โx)5 โ 5 = (โx)0 = 1 โ7z (โ7z)5 = (โ7z)1 (โ7z)5 = (โ7z)1 โ 5 = (โ7z)โ4 = 1 (โ7z)4 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 41 1.19 Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents. a. b. t โ11 โ
t 6 2512 2513 Finding the Power of a Product To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider (pq)3. We begin by using the associative and commutative properties of multiplication to regroup the factors. 3 factors (pq)3 = (pq) โ
(pq) โ
(pq factors โ
q ๏ฟฝ
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๏ฟฝ๏ฟฝ q โ
q 3 factors = p โ
p โ
p = p3 โ
q3 In other words, (pq)3 = p3 โ
q3. The Power of a Product Rule of Exponents For any real numbers a and b and any integer n, the power of a product rule of exponents states that (ab) n = an bn (1.6) Example 1.20 Using the Power of a Product Rule Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents. a. b. c. d. e. 3 โ โab2โ โ (2t)15 3 โโ2w3โ โ โ 1 (โ7z)4 7 โeโ2 f 2โ โ โ Solution Use the product and quotient rules and the new definitions to simplify each expression. 42 Chapter 1 Prerequisites a. b. c. d. e. 3 โab2โ โ โ = (a)3 โ
โb2โ โ โ 3 = a1 โ
3 โ
b2 โ
3 = a3 b6 2t 15 = (2)15 โ
(t)15 = 215 t 15 = 32, 768t 15 3 โโ2w3โ โ โ = (โ2)3 โ
โw3โ โ โ 3 = โ8 โ
w3 โ
3 = โ8w9 1 (โ7z)4 = 1 (โ7)4 โ
(z)4 = 1 2, 401z4 โeโ2 f 2โ โ โ 7 โeโ2โ โ โ = 7 7 โ f 2โ โ โ โ
= eโโ14 f 14 = f 14 e14 Simplify each of the following products as much as possible using the power of a product rule. Write 1.20 answers with positive exponents. a. b. c. d. 5 โg2 h3โ ๏ฟฝ
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๏ฟฝ๏ฟฝ โ (5t)3 3 โโ3y5โ โ โ 1 โa6 b7โ โ โ 3 e. 4 โr 3 sโ2โ โ โ Finding the Power of a Quotient To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, letโs look at the following example. Letโs rewrite the original problem differently and look at the result. โeโ2 f 2โ โ โ 7 = f 14 e14 7 โeโ2 f 2 โ e2 โ f 14 e14 = = It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 43 7 โeโ2 f 2โ โ โ 7 โ โ f 2 โ โ e2 โ โ ( f 2)7 (e2)7 f 2 โ
7 e2 โ
7 f 14 e14 = = = = The Power of a Quotient Rule of Exponents For any real numbers a and b and any integer n, the power of a quotient rule of exponents states that n โ โ a b โ โ = an bn (1.7) Example 1.21 Using the Power of a Quotient Rule Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents. 3 โ โ 6 a. โ 4 โ z11 b. โ โ โ p q3 โ โ โ c. โ โ1 โ t 2 27 โ โ d. e. 4 โj3 k โ2โ โ โ 3 โ โmโ2 n
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โ2โ โ Solution a. โ 4 โ z11 3 โ โ = (4)3 โz11โ โ โ 3 = 64 z11 โ
3 = 64 z33 b. 6 โ โ โ p q3 โ โ โ = (p)6 โq3โ โ โ 6 = p1 โ
6 q3 โ
6 = p6 q18 44 Chapter 1 Prerequisites c. โ โ1 โ t 2 27 โ โ = (โ1)27 โt 2โ โ โ 27 = โ1 t 2 โ
27 = โ1 t 54 = โ 1 t 54 d. โj3 k โ2โ โ โ 4 = 4 โ โ โ j3 j3โ โ โ โk 2โ โ โ j3 โ
4 k 2 โ
4 = j12 k 8 e. โ โmโ2 nโ2โ โ 3 = โ 1 โ m2 n2 3 โ โ = (1)3 โ โm2 n2โ โ 3 = 1 3 โ โn2โ โ 3 = โ โm2โ โ 1 m2 โ
3 โ
n2 โ
3 = 1 m6 n6 Simplify each of the following quotients as much as possible using the power of a quotient rule. Write 1.21 answers with positive exponents. a. 3 b5 c โ โ โ โ b. โ 5 โ u8 4 โ โ c. โ โ1 โ w3 35 โ โ d. e. 8 โpโ4 q3โ โ โ 4 โcโ5 d โ3โ โ โ Simplifying Exponential Expressions Recall that
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to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions. Example 1.22 Simplifying Exponential Expressions Simplify each expression and write the answer with positive exponents only. a. b. 3 โ6m2 nโ1โ โ โ 175 โ
17โ4 โ
17โ3 c. โ uโ1 v โ vโ1 2 โ โ This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 45 d. e. f. โโ2a3 bโ1โ โ โ โ5aโ2 b2โ โ โ โ โx2 2 โ โ 4 โ โ โx2 2 โ โ4 5 โ3w2โ โ โ โ6wโ2โ โ โ 2 Solution a. 3 โ โ6m2 nโ1โ โ 3 3 โ โnโ1โ โm2โ = (6)3 โ โ โ = 63 m2 โ
3 nโ1 โ
3 = 216m6 nโ3 = 216m6 n3 175 โ
17โ4 โ
17โ3 = 175 โ 4 โ 3 = 17โ2 = 1 172 or 1 289 โ uโ1 v โ vโ1 2 โ โ = = (uโ1 v)2 (vโ1)2 uโ2 v2 vโ2 = uโ2 v2 โ (โ2) = uโ2 v4 v4 u2 = b. c. d. e. The power of a product rule The power rule Simplify. The negative exponent rule The product rule Simplify. The negative exponent rule The power of a quotient rule The power of a product rule The quotient rule Simplify. The negative exponent rule โ โ โโ2a3
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bโ1โ โ5aโ2 b2โ โ โ = โ2 โ
5 โ
a3 โ
aโ2 โ
bโ1 โ
b2 = โ10 โ
a3 โ 2 โ
bโ1 + 2 = โ10ab โ โx2 2 โ โ 4 โ โ โx2 2 โ โ4 = โ โ โx2 2 โ 4 โ 4 = โ โ โx2 2 โ 0 = 1 Commutative and associative laws of multiplication The product rule Simplify. The product rule Simplify. The zero exponent rule 46 Chapter 1 Prerequisites f. (3w2)5 (6wโ2)2 = (3)5 โ
(w2)5 (6)2 โ
(wโ2)2 = 35 w2 โ
5 62 wโ2 โ
2 = 243w10 36wโ4 = 27w10 โ (โ4) 4 = 27w14 4 The power of a product rule The power rule Simplify. The quotient rule and reduce fraction Simplify. 1.22 Simplify each expression and write the answer with positive exponents only. a. b. c. d. e. f. โ3 โ โ2uvโ2โ โ x8 โ
xโ12 โ
x 2 โ โ โ e2 f โ3 f โ1 โ โ โ โ โ9r โ5 s3โ โ3r 6 sโ4โ โ โ โ twโ2โ โ โ โ 4 9 โ3 โ โ 4 9 3 twโ2โ โ 4 โ โ2h2 kโ โ โ โ7hโ1 k 2โ โ 2 Using Scientific Notation Recall at the beginning of the section that we found the number 1.3 ร 1013 when describing bits of information in digital images. Other extreme numbers include the width of a
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human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers such as these? A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raised to a power of n. If you moved the decimal left as in a very large number, n is positive. If you moved the decimal right as in a small large number, n is negative. For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 47 We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number. 2.780418 ร 106 Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right. Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number. 4.7 ร 10โ13 Scientific Notation A number is written in scientific notation if it is written in the form a ร 10 n, where 1 โค |a| < 10 and n is an integer. Example 1.23 Converting Standard Notation to Scientific Notation Write each number in scientific notation. a. Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m b. Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m c. Number of stars in Andromeda Galaxy
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: 1,000,000,000,000 d. Diameter of electron: 0.00000000000094 m e. Probability of being struck by lightning in any single year: 0.00000143 Solution a. b. c. 24,000,000,000,000,000,000,000 m 24,000,000,000,000,000,000,000 m โ 22 places 2.4 ร 1022 m 1,300,000,000,000,000,000,000 m 1,300,000,000,000,000,000,000 m โ 21 places 1.3 ร 1021 m 1,000,000,000,000 1,000,000,000,000 โ 12 places 1 ร 1012 48 Chapter 1 Prerequisites d. e. 0.00000000000094 m 0.00000000000094 m โ 6 places 9.4 ร 10โ13 m 0.00000143 0.00000143 โ 6 places 1.43 ร 10โ6 Analysis Observe that, if the given number is greater than 1, as in examples aโc, the exponent of 10 is positive; and if the number is less than 1, as in examples dโe, the exponent is negative. 1.23 Write each number in scientific notation. a. U.S. national debt per taxpayer (April 2014): $152,000 b. World population (April 2014): 7,158,000,000 c. World gross national income (April 2014): $85,500,000,000,000 d. Time for light to travel 1 m: 0.00000000334 s e. Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715 Converting from Scientific to Standard Notation To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal n places to the right if n is positive or n places to the left if n is negative and add zeros as needed. Remember, if n is positive, the value of the number is greater than 1, and if n is negative, the value of the number is less than one. Example 1.24 Converting Scientific Notation to Standard Notation Convert each number in scientific notation to standard notation. a. 3.547 ร 1014 b. โ2 ร 106 c. 7.91 ร 10โ7 d. โ8.05 ร 10โ12 Solution This content is available for free
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at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 49 a. b. c. d. 3.547 ร 1014 3.54700000000000 โ 14 places 354,700,000,000,000 โ2 ร 106 โ2.000000 โ 6 places โ2,000,000 7.91 ร 10โ7 0000007.91 โ 7 places 0.000000791 โ8.05 ร 10โ12 โ000000000008.05 โ 12 places โ0.00000000000805 1.24 Convert each number in scientific notation to standard notation. a. 7.03 ร 105 b. โ8.16 ร 1011 c. โ3.9 ร 10โ13 d. 8 ร 10โ6 Using Scientific Notation in Applications Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around 1.32 ร 1021 molecules of water and 1 L of water holds about 1.22 ร 104 average drops. Therefore, there are approximately โ โ 4.83 ร 1025 atoms in 1 L of water. We simply multiply the decimal terms and add the 3 โ
exponents. Imagine having to perform the calculation without using scientific notation! โ1.32 ร 1021โ โ โ โ
โ1.22 ร 104โ โ When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product โ โ = 35 ร 1010. The answer is not in proper scientific notation because โ5 ร 106โ โ 35 is greater than 10. Consider 35 as 3.5 ร 10. That adds a ten to the exponent of the answer. โ7 ร 104โ โ โ
(35) ร 1010 = (3.5 ร 10) ร 1010 = 3.5 ร โ โ10 ร 1010โ โ = 3.5 ร 1011 50 Chapter 1 Prerequisites Example 1.25 Using Scientific Notation Perform the operations and write the answer in scientific notation. a. b
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. c. d. e. โ6.5 ร 1010โ โ โ8.14 ร 10โ7โ โ โ โ โ4 ร 105โ โ โ รท โ โโ1.52 ร 109โ โ โ2.7 ร 105โ โ โ6.04 ร 1013โ โ โ โ โ1.2 ร 108โ โ โ รท โ โ9.6 ร 105โ โ โ5.62 ร 105โ โ โ โ3.33 ร 104โ โโ1.05 ร 107โ โ โ โ โ Solution a. b. c. d. e. โ6.5 ร 1010โ โ โ8.14 ร 10โ7โ โ โ โ10โ7 ร 1010โ โ โ = (8.14 ร 6.5) โ โ103โ โ = (52.91) โ = 5.291 ร 104 โ4 ร 105โ โ โ รท โ โโ1.52 ร 109โ โ = โ โ 4 โ1.52 โ โ 105 โ โ โ โ 109 โ โ10โ4โ โ (โ2.63) โ = โ2.63 ร 10โ4 โ2.7 ร 105โ โ โ6.04 ร 1013โ โ โ โ105 ร 1013โ โ โ = (2.7 ร 6.04) โ โ1018โ โ = (16.308) โ = 1.6308 ร 1019 โ1.2 ร 108โ โ โ รท ๏ฟฝ
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๏ฟฝ โ9.6 ร 105โ โ = โ โ 1.2 9.6 โ โ 108 โ โ โ โ 105 โ103โ โ = (0.125) โ = 1.25 ร 102 Commutative and associative properties of multiplication Product rule of exponents Scientific n tation Commutative and associative properties of multiplication Quotient rule of exponents Scientific n tation Commutative and associative properties of multiplication Product rule of exponents Scientific n tation Commutative and associative properties of multiplication Quotient rule of exponents Scientific n tation โ5.62 ร 105โ โ โ3.33 ร 104โ โ โโ1.05 ร 107โ โ โ โ โ104 ร 107 ร 105โ โ โ = [3.33 ร (โ1.05) ร 5.62] โ โ1016โ โ โ (โ19.65) โ = โ1.965 ร 1017 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 51 1.25 Perform the operations and write the answer in scientific notation. a. b. c. d. e. โโ7.5 ร 108โ โ โ โ1.13 ร 10โ2โ โ โ โ1.24 ร 1011โ โ โ รท โ1.55 ร 1018โ โ โ โ โ3.72 ร 109โ โ8 ร 103โ โ โ โ โ9.933 ร 1023โ โ โ รท โ โโ2.31 ร 1017โ โ โโ6.04 ร 109โ โ โโ2.81 ร 102โ โ โ7.3 ร 102โ โ โ ๏ฟฝ
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๏ฟฝ โ Example 1.26 Applying Scientific Notation to Solve Problems In April 2014, the population of the United States was about 308,000,000 people. The national debt was about $17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations. Solution The population was 308,000,000 = 3.08 ร 108. The national debt was $17,547,000,000,000 โ $1.75 ร 1013. To find the amount of debt per citizen, divide the national debt by the number of citizens. โ1.75 ร 1013โ โ โ รท โ โ3.08 ร 108โ โ = โ โ โ โ โ
โ 1013 1.75 โ 108 3.08 โ 0.57 ร 105 = 5.7 ร 104 โ โ The debt per citizen at the time was about $5.7 ร 104, or $57,000. 1.26 An average human body contains around 30,000,000,000,000 red blood cells. Each cell measures approximately 0.000008 m long. Write each number in scientific notation and find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations. Access these online resources for additional instruction and practice with exponents and scientific notation. โข Exponential Notation (http://openstaxcollege.org/l/exponnot) โข Properties of Exponents (http://openstaxcollege.org/l/exponprops) โข Zero Exponent (http://openstaxcollege.org/l/zeroexponent) โข Simplify Exponent Expressions (http://openstaxcollege.org/l/exponexpres) โข Quotient Rule for Exponents (http://openstaxcollege.org/l/quotofexpon) โข Scientific Notation (http://openstaxcollege.org/l/scientificnota) โข Converting to Decimal Notation (http://openstaxcollege.org/l/decimalnota) 52 Chapter 1 Prerequisites 1.2 EXERC
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ISES Verbal 69. Is 23 the same as 32? Explain. 70. When can you add two exponents? 5 โ33 รท 34โ โ โ the following exercises, express the decimal For scientific notation. in 71. What is the purpose of scientific notation? 72. Explain what a negative exponent does. 89. 0.0000314 90. 148,000,000 Numeric For the following exercises, simplify the given expression. Write answers with positive exponents. 73. 92 74. 15โ2 75. 76. 77. 32 ร 33 44 รท 4 โ2 โ โ22โ โ 78. (5 โ 8)0 79. 80. 81. 113 รท 114 65 ร 6โ7 2 โ80โ โ โ 82. 5โ2 รท 52 For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents. 42 ร 43 รท 4โ4 612 69 โ โ123 ร 12 โ โ 10 106 รท โ1010โ โ โ โ2 7โ6 ร 7โ3 83. 84. 85. 86. 87. 88. This content is available for free at https://cnx.org/content/col11758/1.5 For the following exercises, convert each number in scientific notation to standard notation. 91. 92. 1.6 ร 1010 9.8 ร 10โ9 Algebraic For the following exercises, simplify the given expression. Write answers with positive exponents. 93. 94. a3 a2 a mn2 mโ2 95. 2 โb3 c4โ โ โ 96. 97. 98. 99. โ5 โ โxโ3 y2 โ โ โ โ ab2 รท d โ3 โ1 โw0 x5โ โ โ m4 n0 100. 2 yโ4 โ โy2โ โ 101. pโ4 q2 p2 qโ3 102. (l ร w)2 103. 3 โy7โ โ โ รท x14 104. 53
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โ 123 m33 โ 4โ3 2 โ โ 120. 173 รท 152 x3 Extensions For the following exercises, simplify the given expression. Write answers with positive exponents. 121. 122. 123. 124. 125. 2 โ2 โ โ โ โ โ 32 โ a3 โ โ a4 22 โ โ โ62 โ24 โ โ5 2 โ โ x y โ โ รท m2 n3 a2 cโ3 โ
aโ7 nโ2 m2 c4 10 โ โ โ x6 y3 x3 yโ3 โ
yโ7 xโ3 โ โ โ โ3 โ โ โ โ โ โab2 cโ โ bโ3 2 โ โ โ โ Avogadroโs constant is used to calculate the number 126. of particles in a mole. A mole is a basic unit in chemistry to is measure the amount of a substance. The constant 6.0221413 ร 1023. Write Avogadroโs in standard notation. constant Planckโs constant is an important unit of measure in 127. quantum physics. It describes the relationship between energy and frequency. The constant is written as 6.62606957 ร 10โ34. Write Planckโs constant in standard notation. Chapter 1 Prerequisites 2 โ โ โ a โ 23 105. 52 m รท 50 m 106. 107. 108. (16 x)2 yโ1 23 (3a)โ2 2 โma6โ โ โ 1 m3 a2 109. 3 โbโ3 cโ โ โ 110. โx2 y13 รท y0โ โ โ 2 111. โ9z3โ โ โ โ2 y Real-World Applications To reach escape velocity, a rocket must travel at the 112. rate of 2.2 ร 106 ft/min. Rewrite the rate in standard notation. A dime
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is the thinnest coin in U.S. currency. A dimeโs 113. thickness measures 2.2 ร 106 m. Rewrite the number in standard notation. The average distance between Earth and the Sun is 114. 92,960,000 mi. Rewrite the distance using scientific notation. A 115. 1,099,500,000,000 bytes. Rewrite in scientific notation. terabyte approximately made of is The Gross Domestic Product (GDP) for the United 116. States in the first quarter of 2014 was $1.71496 ร 1013. Rewrite the GDP in standard notation. 117. One picometer is approximately 3.397 ร 10โ11 in. Rewrite this length using standard notation. The value of the services sector of the U.S. economy 118. in the first quarter of 2012 was $10,633.6 billion. Rewrite this amount in scientific notation. Technology For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth. 119. 54 Chapter 1 Prerequisites 1.3 | Radicals and Rational Expressions Learning Objectives In this section students will: 1.3.1 Evaluate square roots. 1.3.2 Use the product rule to simplify square roots. 1.3.3 Use the quotient rule to simplify square roots. 1.3.4 Add and subtract square roots. 1.3.5 Rationalize denominators. 1.3.6 Use rational roots. A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in Figure 1.6, and use the Pythagorean Theorem. Figure 1.6 a2 + b2 = c2 52 + 122 = c2 169 = c2 Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one. Evaluating Square Roots When the square root of a number is squared, the result is the original number. Since 42 = 16, the square root of 16 is 4. The square root function is the inverse of the
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squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root. In general terms, if a is a positive real number, then the square root of a is a number that, when multiplied by itself, gives a. The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals a. The square root obtained using a calculator is the principal square root. The principal square root of a is written as a. The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 55 Principal Square Root The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: a. Does 25 = ยฑ 5? No. Although both 52 and (โ5)2 are 25, square root. The principal square root of 25 is 25 = 5. the radical symbol implies only a nonnegative root, the principal Example 1.27 Evaluating Square Roots Evaluate each expression. a. b. c. d. 100 16 25 + 144 49 โ 81 Solution a. b. c. d. 100 = 10 because 102 = 100 16 = 4 = 2 because 42 = 16 and 22 = 4 25 + 144 = 169 = 13 because 132 = 169 49 โ 81 = 7 โ 9 = โ2 because 72 = 49 and 92 = 81 For 25 + 144, can we find the square roots before adding? No. 25 + 144 = 5 + 12 = 17. This is not equivalent to 25 + 144 = 13. The order of operations requires us to add the terms in the radicand before finding the square root. 1.27 Evaluate each expression. a. b. c. d. 225 81 25 โ 9 36 + 121 56 Chapter 1 Prerequisites Using the Product Rule to Simplify Square Roots To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root
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of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite 15 as 3 โ
5. We can also use the product rule to express the product of multiple radical expressions as a single radical expression. The Product Rule for Simplifying Square Roots If a and b are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b. ab = a โ
b Given a square root radical expression, use the product rule to simplify it. 1. Factor any perfect squares from the radicand. 2. Write the radical expression as a product of radical expressions. 3. Simplify. Example 1.28 Using the Product Rule to Simplify Square Roots Simplify the radical expression. a. b. 300 162a5 b4 Solution a. b. 100 โ
3 100 โ
3 10 3 Factor perfect square from radicand. Write radical expression as product of radical expressions. Simplify. 81a4 b4 โ
2a 81a4 b4 โ
2a 9a2 b2 2a Factor perfect square from radicand. Write radical expression as product of radical expressions. Simplify. 1.28 Simplify 50x2 y3 z. Given the product of multiple radical expressions, use the product rule to combine them into one radical expression. 1. Express the product of multiple radical expressions as a single radical expression. 2. Simplify. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 57 Example 1.29 Using the Product Rule to Simplify the Product of Multiple Square Roots Simplify the radical expression. 12 โ
3 Solution 12 โ
3 Express the product as a single radical expression. 36 6 Simplify. 1.29 Simplify 50x โ
2x assuming x > 0. Using the Quotient Rule to Simplify Square Roots Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite 5 2. as 5 2 The Quotient Rule for Simplifying Square Roots The square root of the quotient
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a b is equal to the quotient of the square roots of a and b, where b โ 0. a b = a b Given a radical expression, use the quotient rule to simplify it. 1. Write the radical expression as the quotient of two radical expressions. 2. Simplify the numerator and denominator. Example 1.30 Using the Quotient Rule to Simplify Square Roots Simplify the radical expression. 5 36 Solution 58 Chapter 1 Prerequisites 5 36 5 6 Write as quotient of two radical expressions. Simplify denominator. 1.30 Simplify 2x2 9y4. Example 1.31 Using the Quotient Rule to Simplify an Expression with Two Square Roots Simplify the radical expression. 234x11 y 26x7 y Solution 234x11 y 26x7 y 9x4 3x2 Combine numerator and denominator into one radical expression. Simplify fraction. Simplify square root. 1.31 Simplify 9a5 b14 3a4 b5. Adding and Subtracting Square Roots We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of 2 and 3 2 is 4 2. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression 18 can be written with a 2 in the radicand, as 3 2, so 2 + 18 = 2 + 3 2 = 4 2. Given a radical expression requiring addition or subtraction of square roots, solve. 1. Simplify each radical expression. 2. Add or subtract expressions with equal radicands. Example 1.32 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 59 Adding Square Roots Add 5 12 + 2 3. Solution We can rewrite 5 12 as 5 4 ยท 3. According the product rule, this becomes 5 4 3. The square root of 4 is 2, so the expression becomes 5(2) 3, which is 10 3. Now we can the terms have the same radicand so we can add. 10 3 + 2 3 = 12 3 1.32 Add 5 + 6 20. Example 1.33 Subtracting Square Roots Subtract 20 72a3 b4 c โ 14 8a3 b4 c. Solution Rewrite each term so they have equal radic
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ands. 2 c โb2โ โ 20 72a3 b4 c = 20 9 4 2 a a2 โ = 20(3)(2)|a|b2 2ac = 120|a|b2 2ac 14 8a3 b4 c = 14 2 4 a a2 โ = 14(2)|a|b2 2ac = 28|a|b2 2ac 2 c โb2โ โ Now the terms have the same radicand so we can subtract. 120|a|b2 2ac โ 28|a|b2 2ac = 92|a|b2 2ac 1.33 Subtract 3 80x โ 4 45x. Rationalizing Denominators When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator. 60 Chapter 1 Prerequisites We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical. For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is b c, multiply by c c. For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the then the conjugate is a โ b c. denominator. If the denominator is a + b c, Given an expression with a single square root radical term in the denominator, rationalize the denominator. a. Multiply the numerator and denominator by the radical in the denominator. b. Simplify. Example 1.34 Rationalizing a Denominator Containing a Single Term in simplest form. Write 2 3 3 10 Solution The radical in the denominator is 10. So multiply the fraction by 10 10. Then simplify. โ
10 10 2 3 3 10 2 30 30 30 15 1.34 Write 12 3 2 in simplest form. Given an expression with a radical term and a constant in the denominator, rationalize the denominator.
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1. Find the conjugate of the denominator. 2. Multiply the numerator and denominator by the conjugate. 3. Use the distributive property. 4. Simplify. Example 1.35 Rationalizing a Denominator Containing Two Terms This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 61 Write 4 1 + 5 in simplest form. Solution Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of 1 + 5 is 1 โ 5. Then multiply the fraction by 4 5 โ 1 Use the distributive property. Simplify. 1.35 Write 7 2 + 3 in simplest form. Using Rational Roots Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number. Understanding nth Roots Suppose we know that a3 = 8. We want to find what number raised to the 3rd power is equal to 8. Since 23 = 8, we say that 2 is the cube root of 8. The nth root of a is a number that, when raised to the nth power, gives a. For example, โ3 is the 5th root of โ243 because (โ3)5 = โ243. If a is a real number with at least one nth root, then the principal nth root of a is the number with the same sign as a that, when raised to the nth power, equals a. The principal nth root of a is written as an expression, n is called the index of the radical., where n is a positive integer greater than or equal to 2. In the radical Principal nth Root If a is a real number with at least one nth root, then the principal nth root of a, written as an, the same sign as a that, when raised to the nth power, equals a. The index of the radical is n. is the number with Example 1.36 Simplifying nth Roots Simplify each of the following: a. 5 โ32 62 Chapter 1 Prerequisites b. 4 4 4 โ
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1, 024 3 c. โ 8x6 125 d. 4 8 3 4 โ 48 Solution 5 a. โ32 = โ2 because (โ2)5 = โ32 b. First, express the product as a single radical expression. 4,096 4 = 8 because 84 = 4,096 c. d. 3 โ 8x6 3 125 โ2x2 Write as quotient of two radical expressions. Simplify. Simplify to get equal radicands. Add. 1.36 Simplify. a. b. c. 3 โ216 4 3 80 4 5 3 6 9, 000 3 + 7 576 Using Rational Exponents Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index n is even, then a cannot be negative. = an We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root. a 1 n All of the properties of exponents that we learned for integer exponents also hold for rational exponents. m n a = ( an ) m = amn Rational Exponents Rational exponents are another way to express principal nth roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is = amn = ( an ) m n a m This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 63 Given an expression with a rational exponent, write the expression as a radical. 1. Determine the power by looking at the numerator of the exponent. 2. Determine the root by looking at the denominator of the exponent. 3. Using the base as the radicand, raise the radicand to the power and use the root as the index. Example 1.37 Writing Rational Exponents as Radicals Write 343 2 3 as a radical. Simplify. Solution The 2 tells us the power and the 3 tells us the root. 2 3 = 2 โ 3 โ 343 โ โ 343 3 = 3432 3 We know that 343 root before squaring for this problem. In general, it is easier to find the root first and then
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raise it to a power. = 7 because 73 = 343. Because the cube root is easy to find, it is easiest to find the cube 2 3 = 2 โ 3 โ 343 โ โ 343 = 72 = 49 1.37 5 2 as a radical. Simplify. Write 9 Example 1.38 Writing Radicals as Rational Exponents using a rational exponent. Write 4 a2 7 Solution The power is 2 and the root is 7, so the rational exponent will be 2 7. We get 4 2 7 a. Using properties of exponents, โ2 7. = 4a we get 4 a2 7 1.38 Write x (5y)9 using a rational exponent. 64 Chapter 1 Prerequisites Example 1.39 Simplifying Rational Exponents Simplify: a. b. โ โ2x โ 3 4 โ โ โ3x โ โ โ 1 5 โ โ โ 5 โ 1 2 โ โ 16 9 โ โ Solution a. 30x 30x 30x b Multiply the coefficien. Use properties of exponents. 19 20 Simplify. 1 2 โ โ โ โ 9 16 9 16 9 16 3 4 Use definition of ne ative exponents. Rewrite as a radical. Use the quotient rule. Simplify. 1.39 Simplify (8x) 1 3 โ โ14x โ 6 5 โ โ. โ Access these online resources for additional instruction and practice with radicals and rational exponents. โข Radicals (http://openstaxcollege.org/l/introradical) โข Rational Exponents (http://openstaxcollege.org/l/rationexpon) โข Simplify Radicals (http://openstaxcollege.org/l/simpradical) โข Rationalize Denominator (http://openstaxcollege.org/l/rationdenom) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 65 1.3 EXERCISES Verbal What does it mean when a radical does not have an 128. index? Is the expression equal to the radicand? Explain. Where would radicals come in the order
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of 129. operations? Explain why. Every number will have two square roots. What is the 130. principal square root? Can a radical with a negative radicand have a real 131. square root? Why or why not? Numeric For the following exercises, simplify each expression. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 256 256 4(9 + 16) 289 โ 121 196 1 98 27 64 81 5 800 169 + 144 8 50 18 162 192 14 6 โ 6 24 15 5 + 7 45 150 149. 150. 151. 152. 153. 154. 155. 156. 96 100 ( 42)โ โ 30โ โ 12 3 โ 4 75 4 225 405 324 360 361 5 1 + 3 8 1 โ 17 157. 4 16 158. 3 128 3 + 3 2 159. 5 โ32 243 160. 4 15 125 4 5 161. 3 3 โ432 3 + 16 Algebraic For the following exercises, simplify each expression. 162. 163. 400x4 4y2 164. 49p 165. 1 2 โ144p2 q6โ โ โ 166. 5 2 289 m 167. 9 3m2 + 27 66 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 3 ab2 โ b a 4 2n 16n4 225x3 49x 3 44z + 99z 50y8 490bc2 32 14d 3 2 63p q 8 1 โ 3x 20 121d 4 3 2 32 โ w 3 2 50 w 108x4 + 27x4 12x 2 + 2 3 147k 3 125n10 42q 36q3 81m 361m2 72c โ 2 2c 144 324d 2 187. 3 24x6 3 + 81x6 188. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 4 162x6 16x4 189. 3 64y 190. 3 128z3 3 โ โ16z3 191. 5 1,024c10 Real-World Applications 192. A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon
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to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating 90,000 + 160,000. What is the length of the guy wire? 193. A car accelerates at a rate of 6 โ 4 t m/s2 where t is the time in seconds after the car moves from rest. Simplify the expression. Extensions For the following exercises, simplify each expression. 194. 195. 196. 1 2 โ 2 8 โ 16 4 โ 2 4 3 2 3 2 โ 16 1 3 8 mn3 a2 cโ3 โ
aโ7 nโ2 m2 c4 197. a a โ c 198. 199. 200. x 64y + 4 y 128y โ โ 250x2 100b3 โ โ โโ โ โ 7 b 125x โ โ 3 4 + 256 64 64 + 256 Chapter 1 Prerequisites 67 1.4 | Polynomials Learning Objectives In this section students will: 1.4.1 Identify the degree and leading coefficient of polynomials. 1.4.2 Add and subtract polynomials. 1.4.3 Multiply polynomials. 1.4.4 Use FOIL to multiply binomials. 1.4.5 Perform operations with polynomials of several variables. Earl is building a doghouse, whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which the dog can enter and exit the house. Earl wants to find the area of the front of the doghouse so that he can purchase the correct amount of paint. Using the measurements of the front of the house, shown in Figure 1.7, we can create an expression that combines several variable terms, allowing us to solve this problem and others like it. Figure 1.7 First find the area of the square in square feet. Then find the area of the triangle in square feet. A = s2 = (2x)2 = 4x2 bh 2x) โ โ โ 3 2 x Next find the
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area of the rectangular door in square feet. A = lw = x โ
1 = x The area of the front of the doghouse can be found by adding the areas of the square and the triangle, and then subtracting the area of the rectangle. When we do this, we get 4x2 + 3 2 x โ x ft2, or 4x2 + 1 2 x ft2. In this section, we will examine expressions such as this one, which combine several variable terms. 68 Chapter 1 Prerequisites Identifying the Degree and Leading Coefficient of Polynomials The formula just found is an example of a polynomial, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as 384ฯ, is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product ai xi is a term of a polynomial. If a term does not contain a variable, it is called a constant., such as 384ฯw, A polynomial containing only one term, such as 5x4, 2x โ 9, is called a binomial. A polynomial containing three terms, such as โ3x2 + 8x โ 7, is called a trinomial. is called a monomial. A polynomial containing two terms, such as We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form. Polynomials A polynomial is an expression that can be written in the form an xn +... + a2 x2 + a1 x + a0 Each real number ai is called a coefficient. The number a0 that is not multiplied by a variable is called a constant. Each product ai xi is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called the degree of a polynomial. The leading term is the term with the highest power, and its coefficient is called the leading coefficient. Given a polynomial
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expression, identify the degree and leading coefficient. 1. Find the highest power of x to determine the degree. 2. 3. Identify the term containing the highest power of x to find the leading term. Identify the coefficient of the leading term. Example 1.40 Identifying the Degree and Leading Coefficient of a Polynomial For the following polynomials, identify the degree, the leading term, and the leading coefficient. a. b. c. 3 + 2x2 โ 4x3 5t 5 โ 2t 3 + 7t 6p โ p3 โ 2 Solution This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 69 a. The highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, โ4x3. The leading coefficient is the coefficient of that term, โ4. b. The highest power of t is 5, so the degree is 5. The leading term is the term containing that degree, 5t 5. The leading coefficient is the coefficient of that term, 5. c. The highest power of p is 3, so the degree is 3. The leading term is the term containing that degree, โ p3, The leading coefficient is the coefficient of that term, โ1. 1.40 Identify the degree, leading term, and leading coefficient of the polynomial 4x2 โ x6 + 2x โ 6. Adding and Subtracting Polynomials We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, 5x2 and โ2x2 are like terms, and can be added to get 3x2, but 3x and 3x2 are not like terms, and therefore cannot be added. Given multiple polynomials, add or subtract them to simplify the expressions. 1. Combine like terms. 2. Simplify and write in standard form. Example 1.41 Adding Polynomials Find the sum. โ โ โ12x2 + 9x โ 21 โ + โ โ โ4x3 + 8x2 โ 5x + 20 โ Solution โ12x2 + 8x2โ โ 4x3 + 4x3 + 20x2 + 4x
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โ 1 โ + (9x โ 5x) + (โ21 + 20) Combine like terms. Simplify. Analysis We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not. 1.41 Find the sum. โ โ โ2x3 + 5x2 โ x + 1 โ + โ โ โ2x2 โ 3x โ 4 โ 70 Chapter 1 Prerequisites Example 1.42 Subtracting Polynomials Find the difference. โ โ โ7x4 โ x2 + 6x + 1 โ โ โ โ โ5x3 โ 2x2 + 3x + 2 โ Solution โโx2 + 2x2โ โ 7x4 โ 5x3 + 7x4 โ 5x3 + x2 + 3x โ 1 โ + (6x โ 3x) + (1 โ 2) Combine like terms. Simplify. Analysis Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first. 1.42 Find the difference. โ โ โโ7x3 โ 7x2 + 6x โ 2 โ โ โ โ โ4x3 โ 6x2 โ x + 7 โ Multiplying Polynomials Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials. Multiplying Polynomials Using the Distributive Property To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each
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term of the polynomial. We can distribute the 2 in 2(x + 7) to obtain the equivalent expression 2x + 14. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify. Given the multiplication of two polynomials, use the distributive property to simplify the expression. 1. Multiply each term of the first polynomial by each term of the second. 2. Combine like terms. 3. Simplify. Example 1.43 Multiplying Polynomials Using the Distributive Property Find the product. โ โ โ3x2 โ x + 4 (2x + 1) โ Solution This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 71 2xโ โ โ โ โ3x2 โ x + 4 โ3x2 โ x + 4 โ โ + 1 โ6x3 โ 2x2 + 8xโ โ โ โ โ3x2 โ x + 4 โ + โ โโ2x2 + 3x2โ โ 6x3 + 6x3 + x2 + 7x + 4 โ + (8x โ x) + 4 Use the distributive property. Multiply. Combine like terms. Simplify. Analysis We can use a table to keep track of our work, as shown in Table 1.2. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify. 3x2 โx 2x 6x3 โ2x2 +1 3x2 โx +4 8x 4 Table 1.2 1.43 Find the product. โ โx3 โ 4x2 + 7 (3x + 2) โ โ Using FOIL to Multiply Binomials A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply
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the first terms, the outer terms, the inner terms, and then the last terms of each binomial. The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms. Given two binomials, use FOIL to simplify the expression. 1. Multiply the first terms of each binomial. 2. Multiply the outer terms of the binomials. 3. Multiply the inner terms of the binomials. 4. Multiply the last terms of each binomial. 5. Add the products. 6. Combine like terms and simplify. 72 Chapter 1 Prerequisites Example 1.44 Using FOIL to Multiply Binomials Use FOIL to find the product. (2x - 10)(3x + 3) Solution Find the product of the first terms. Find the product of the outer terms. Find the product of the inner terms. Find the product of the last terms. 6x2 + 6x โ 54x โ 54 6x2 + (6x โ 54x) โ 54 6x2 โ 48x โ 54 Add the products. Combine like terms. Simplify. 1.44 Use FOIL to find the product. (x + 7)(3x โ 5) Perfect Square Trinomials Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Letโs look at a few perfect square trinomials to familiarize ourselves with the form. (x + 5)2 = x2 + 10x + 25 (x โ 3)2 = x2 โ 6x + 9 (4x โ 1)2 = 4x2 โ 8x + 1 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 73 Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see
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that the first sign of the trinomial is the same as the sign of the binomial. Perfect Square Trinomials When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared. (x + a)2 = (x + a)(x + a) = x2 + 2ax + a2 (1.8) Given a binomial, square it using the formula for perfect square trinomials. 1. Square the first term of the binomial. 2. Square the last term of the binomial. 3. For the middle term of the trinomial, double the product of the two terms. 4. Add and simplify. Example 1.45 Expanding Perfect Squares Expand (3x โ 8)2. Solution Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms. Simplify (3x)2 โ 2(3x)(8) + (โ8)2 9x2 โ 48x + 64. (1.9) 1.45 Expand (4x โ 1)2. Difference of Squares Another special product is called the difference of squares, which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Letโs see what happens when we multiply (x + 1)(x โ 1) using the FOIL method. (x + 1)(x โ 1) = x2 โ x + x โ 1 = x2 โ 1 The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, letโs look at a few examples. 74 Chapter 1 Prerequisites (x + 5)(x โ 5) = x2 โ 25 (x + 11)(x โ 11) = x2 โ 121 (2x + 3)(2x โ 3) = 4x2 โ 9 Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term. Is there a special form for the sum of squares? No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares. Difference of Squares When a binomial is multiplied by a
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binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term. (a + b)(a โ b) = a2 โ b2 (1.10) Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares. 1. Square the first term of the binomials. 2. Square the last term of the binomials. 3. Subtract the square of the last term from the square of the first term. Example 1.46 Multiplying Binomials Resulting in a Difference of Squares Multiply (9x + 4)(9x โ 4). Solution Square the first term to get (9x)2 = 81x2. Square the last term to get 42 = 16. Subtract the square of the last term from the square of the first term to find the product of 81x2 โ 16. 1.46 Multiply (2x + 7)(2x โ 7). Performing Operations with Polynomials of Several Variables We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example: (a + 2b)(4a โ b โ c) a(4a โ b โ c) + 2b(4a โ b โ c) 4a2 โ ab โ ac + 8ab โ 2b2 โ 2bc 4a2 + ( โ ab + 8ab) โ ac โ 2b2 โ 2bc 4a2 + 7ab โ ac โ 2bc โ 2b2 Use the distributive property. Multiply. Combine like terms. Simplify. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 75 Example 1.47 Multiplying Polynomials Containing Several Variables Multiply (x + 4)(3x โ 2y + 5). Solution Follow the same steps that we used to multiply polynomials containing only one variable. x(3x โ 2y + 5) + 4(3x โ 2y + 5) 3x2 โ 2xy + 5x + 12x โ 8y + 20 3x2 โ 2xy + (5x + 12x) โ 8y +
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20 3x2 โ 2xy + 17x โ 8y + 20 Use the distributive property. Multiply. Combine like terms. Simplify. 1.47 Multiply (3x โ 1)(2x + 7y โ 9). Access these online resources for additional instruction and practice with polynomials. โข Adding and Subtracting Polynomials (http://openstaxcollege.org/l/addsubpoly) โข Multiplying Polynomials (http://openstaxcollege.org/l/multiplpoly) โข Special Products of Polynomials (http://openstaxcollege.org/l/specialpolyprod) 76 Chapter 1 Prerequisites 1.4 EXERCISES Verbal 201. Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false. 202. Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial. 203. You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps? 204. State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial. 217. (4x + 2)(6x โ 4) 218. โ2c2 โ 3cโ โ โ14c2 + 4cโ โ โ โ 219. โ โ6b2 โ 6 โ โ โ4b2 โ 4 โ โ โ 220. (3d โ 5)(2d + 9) 221. (9v โ 11)(11v โ 9) 222. โ โ4t 2 + 7tโ โ โ โโ3t 2 + 4 โ โ 223. โ โ โn2 + 9 (8n โ 4) โ For the following exercises, expand the binomial. Algebraic 224. (4x + 5)2 For the following exercises, identify the degree
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of the polynomial. 225. (3y โ 7)2 205. 7x โ 2x2 + 13 206. 207. 208. 14m3 + m2 โ 16m + 8 โ625a8 + 16b4 200p โ 30p2 m + 40m3 209. x2 + 4x + 4 210. 6y4 โ y5 + 3y โ 4 226. (12 โ 4x)2 227. โ โ4p + 9โ โ 2 228. (2m โ 3)2 229. (3y โ 6)2 230. (9b + 1)2 For the following exercises, multiply the binomials. For the following exercises, find the sum or difference. 231. (4c + 1)(4c โ 1) 211. โ12x2 + 3xโ โ โ โ โ โ โ8x2 โ19 โ 212. โ4z3 + 8z2 โ zโ โ โ + โ โโ2z2 + z + 6 โ โ 213. โ โ โ6w2 + 24w + 24 โ โ (3w โ 6w + 3) 232. (9a โ 4)(9a + 4) 233. (15n โ 6)(15n + 6) 234. (25b + 2)(25b โ 2) 235. (4 + 4m)(4 โ 4m) 236. (14p + 7)(14p โ 7) โ โ โ7a3 + 6a2 โ 4a โ 13 โ + โ โ โโ3a3 โ 4a2 + 6a + 17 โ 214. 215. โ โ โ11b4 โ 6b3 + 18b2 โ 4b + 8 โ โ โ3b3 + 6b2 + 3bโ โ โ 237. (11q โ 10)(11q + 10) 216. โ โ โ49p2 โ 25 โ + โ โ โ16p4 โ 32p2 + 16 โ
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For the following exercises, multiply the polynomials. For the following exercises, find the product. 238. โ โ โ2x2 + 2x + 1 โ (4x โ 1) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 77 โa2 โ 4c2โ โ โa2 + 4ac + 4c2โ โ โ โ 239. โ โ โ โ4t 2 โ 1 โ4t 2 + t โ 7 โ โ โ 240. โ โ โx2 โ 2x + 1 (x โ 1) โ 241. โ โy2 โ 4y โ 9 (y โ 2) โ โ 242. โ โ โ6k 2 + 5k โ 1 (6k โ 5) โ 243. โ โ โ3p2 + 2p โ 10 โ (p โ 1) 244. โ โ โ2m2 โ 7m + 9 (4m โ 13) โ 245. (a + b)(a โ b) 246. (4x โ 6y)(6x โ 4y) 247. (4t โ 5u)2 248. (9m + 4n โ 1)(2m + 8) 249. (4t โ x)(t โ x + 1) 250. โ โ โ โa2 + 2ab + b2โ โb2 โ 1 โ โ 251. (4r โ d)(6r + 7d) 252. โ โx2 โ xy + y2โ (x + y) โ Real-World Applications A developer wants to purchase a plot of land to build 253. a house. The area of the plot can be described by the following is expression: (4x + 1)(8x โ 3) where x measured in meters. Multiply the binomials to find the area of the plot in standard form. A prospective buyer wants to know how much
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grain a 254. specific silo can hold. The area of the floor of the silo is (2x + 9)2. The height of the silo is 10x + 10, where x is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold. Extensions For the following exercises, perform the given operations. 255. (4t โ 7)2(2t + 1) โ โ โ โ4t 2 + 2t + 11 โ 256. โ โ โ9b2 โ 36 (3b + 6)(3b โ 6) โ 257. 78 Chapter 1 Prerequisites 1.5 | Factoring Polynomials Learning Objectives In this section students will: 1.5.1 Factor the greatest common factor of a polynomial. 1.5.2 Factor a trinomial. 1.5.3 Factor by grouping. 1.5.4 Factor a perfect square trinomial. 1.5.5 Factor a difference of squares. 1.5.6 Factor the sum and difference of cubes. 1.5.7 Factor expressions using fractional or negative exponents. Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in Figure 1.8. Figure 1.8 The area of the entire region can be found using the formula for the area of a rectangle. A = lw = 10x โ
6x = 60x2 units2 The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of A = s2 = 42 = 16 units2. The other rectangular region has one side of length 10x โ 8 and one side of length 4, giving an area of A = lw = 4(10x โ 8) = 40x โ 32 units2. So the region that must be subtracted has an area of 2(16) + 40x โ 32 = 40x units2. The area of the region that requires grass seed is found by subtracting 60x2 โ 40x units2. This area can also be expressed in factored form as 20x(3x โ 2) units2. We can confirm that this is an equivalent expression by multiplying. Many po
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lynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions. Factoring the Greatest Common Factor of a Polynomial When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, 4 is the GCF of 16 and 20 because it is the largest number that divides evenly into both 16 and 20 The GCF of polynomials works the same way: 4x is the GCF of 16x and 20x2 because it is the largest polynomial that divides evenly into both 16x and 20x2. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 79 When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables. Greatest Common Factor The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials. Given a polynomial expression, factor out the greatest common factor. 1. 2. Identify the GCF of the coefficients. Identify the GCF of the variables. 3. Combine to find the GCF of the expression. 4. Determine what the GCF needs to be multiplied by to obtain each term in the expression. 5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by. Example 1.48 Factoring the Greatest Common Factor Factor 6x3 y3 + 45x2 y2 + 21xy. Solution First, find the GCF of the expression. The GCF of 6, 45, and 21 is 3. The GCF of x3, x2, and x is x. (Note that the GCF of a set of expressions in the form xn will always be the exponent of lowest degree.) And the GCF of y3, y2, and y is y. Combine these to find the GCF of the polynomial, 3xy. Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that 3xyโ โ = 6x3 y3, 3xy(15
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xy) = 45x2 y2, and 3xy(7) = 21xy. โ2x2 y2โ Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by. โ โ โ2x2 y2 + 15xy + 7 (3xy) โ Analysis After factoring, we can check our work by multiplying. Use the distributive property to confirm that โ โ โ = 6x3 y3 + 45x2 y2 + 21xy. โ2x2 y2 + 15xy + 7 (3xy) 1.48 Factor xโ โb2 โ aโ โb2 โ aโ โ โ + 6 โ by pulling out the GCF. Factoring a Trinomial with Leading Coefficient 1 Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial x2 + 5x + 6 has a GCF of 1, but it can be written as the product of the factors (x + 2) and (x + 3). 80 Chapter 1 Prerequisites Trinomials of the form x2 + bx + c can be factored by finding two numbers with a product of c and a sum of b. The trinomial x2 + 10x + 16, for example, can be factored using the numbers 2 and 8 because the product of those numbers is 16 and their sum is 10. The trinomial can be rewritten as the product of (x + 2) and (x + 8). Factoring a Trinomial with Leading Coefficient 1 A trinomial of p + q = b. the form x2 + bx + c can be written in factored form as (x + p)(x + q) where pq = c and Can every trinomial be factored as a product of binomials? No. Some polynomials cannot be factored. These polynomials are said to be prime. Given a trinomial in the form x2 + bx + c, factor it. 1. List factors of c. 2. Find p and q, a pair of factors of c with a sum of b. 3. Write
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the factored expression (x + p)(x + q). Example 1.49 Factoring a Trinomial with Leading Coefficient 1 Factor x2 + 2x โ 15. Solution We have a trinomial with leading coefficient 1, b = 2, and c = โ15. We need to find two numbers with a product of โ15 and a sum of 2. In Table 1.3, we list factors until we find a pair with the desired sum. Factors of โ15 Sum of Factors 1, โ15 โ14 โ1, 15 3, โ5 โ3, 5 Table 1.3 14 โ2 2 Now that we have identified p and q as โ3 and 5, write the factored form as (x โ 3)(x + 5). This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 81 Analysis We can check our work by multiplying. Use FOIL to confirm that (x โ 3)(x + 5) = x2 + 2x โ 15. Does the order of the factors matter? No. Multiplication is commutative, so the order of the factors does not matter. 1.49 Factor x2 โ 7x + 6. Factoring by Grouping Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial 2x2 + 5x + 3 can be rewritten as (2x + 3)(x + 1) using this process. We begin by rewriting the original expression as 2x2 + 2x + 3x + 3 and then factor each portion of the expression to obtain 2x(x + 1) + 3(x + 1). We then pull out the GCF of (x + 1) to find the factored expression. Factor by Grouping To factor a trinomial in the form ax2 + bx + c by grouping, we find two numbers with a product of ac and a sum of b. We use these numbers to divide the x term into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression. Given a trinomial in the form ax2 + bx + c, factor by grouping.
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1. List factors of ac. 2. Find p and q, a pair of factors of ac with a sum of b. 3. Rewrite the original expression as ax2 + px + qx + c. 4. Pull out the GCF of ax2 + px. 5. Pull out the GCF of qx + c. 6. Factor out the GCF of the expression. Example 1.50 Factoring a Trinomial by Grouping Factor 5x2 + 7x โ 6 by grouping. Solution 82 Chapter 1 Prerequisites We have a trinomial with a = 5, b = 7, and c = โ6. First, determine ac = โ30. We need to find two numbers with a product of โ30 and a sum of 7. In Table 1.4, we list factors until we find a pair with the desired sum. Factors of โ30 Sum of Factors 1, โ30 โ1, 30 2, โ15 โ2, 15 3, โ10 โ3, 10 Table 1.4 โ29 29 โ13 13 โ7 7 So p = โ3 and q = 10. 5x2 โ 3x + 10x โ 6 x(5x โ 3) + 2(5x โ 3) (5x โ 3)(x + 2) Rewrite the original expression as ax2 + px + qx + c. Factor out the GCF of each part. Factor out the GCF of the expression. Analysis We can check our work by multiplying. Use FOIL to confirm that (5x โ 3)(x + 2) = 5x2 + 7x โ 6. 1.50 Factor a. 2x2 + 9x + 9 b. 6x2 + x โ 1 Factoring a Perfect Square Trinomial A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term. a2 + 2ab + b2 = (a + b)2 and a2 โ 2ab + b2 = (a โ b)2 We can use this equation to factor any perfect square trinomial. Perfect Square Trinomials A perfect square trinomial can be written as the square of a binomial: This content is available for free at https://cnx.org/content/col
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11758/1.5 Chapter 1 Prerequisites a2 + 2ab + b2 = (a + b)2 Given a perfect square trinomial, factor it into the square of a binomial. 1. Confirm that the first and last term are perfect squares. 2. Confirm that the middle term is twice the product of ab. 3. Write the factored form as (a + b)2. 83 (1.11) Example 1.51 Factoring a Perfect Square Trinomial Factor 25x2 + 20x + 4. Solution Notice that 25x2 and 4 are perfect squares because 25x2 = (5x)2 and 4 = 22. Then check to see if the middle term is twice the product of 5x and 2. The middle term is, indeed, twice the product: 2(5x)(2) = 20x. Therefore, the trinomial is a perfect square trinomial and can be written as (5x + 2)2. 1.51 Factor 49x2 โ 14x + 1. Factoring a Difference of Squares A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied. We can use this equation to factor any differences of squares. a2 โ b2 = (a + b)(a โ b) Differences of Squares A difference of squares can be rewritten as two factors containing the same terms but opposite signs. a2 โ b2 = (a + b)(a โ b) (1.12) Given a difference of squares, factor it into binomials. 1. Confirm that the first and last term are perfect squares. 2. Write the factored form as (a + b)(a โ b). Example 1.52 84 Chapter 1 Prerequisites Factoring a Difference of Squares Factor 9x2 โ 25. Solution Notice that 9x2 and 25 are perfect squares because 9x2 = (3x)2 and 25 = 52. The polynomial represents a difference of squares and can be rewritten as (3x + 5)(3x โ 5). 1.52 Factor 81y2 โ 100. Is there a formula to factor the sum of squares? No. A sum of squares cannot be factored. Factoring the Sum and Difference of Cubes Now, we will look at two new special
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products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial. โ โa2 โ ab + b2โ a3 + b3 = (a + b) โ Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs. โ โa2 + ab + b2โ a3 โ b3 = (a โ b) โ We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: Same Opposite Always Positive. For example, consider the following example. โ x3 โ 23 = (x โ 2) โx2 + 2x + 4 โ โ The sign of the first 2 is the same as the sign between x3 โ 23. The sign of the 2x term is opposite the sign between x3 โ 23. And the sign of the last term, 4, is always positive. Sum and Difference of Cubes We can factor the sum of two cubes as โa2 โ ab + b2โ โ a3 + b3 = (a + b) โ We can factor the difference of two cubes as โa2 + ab + b2โ โ a3 โ b3 = (a โ b) โ (1.13) (1.14) Given a sum of cubes or difference of cubes, factor it. 1. Confirm that the first and last term are cubes, a3 + b3 or a3 โ b3. โa2 โ ab + b2โ โ 2. For a sum of cubes, write the factored form as (a + b) โ . For a difference of cubes, write the โa2 + ab + b2โ โ factored form as (a โ b) โ . This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 85 Example 1.53 Factoring a Sum of Cubes Factor x3 + 512. Solution โ โ Notice that x3 and
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512 are cubes because 83 = 512. Rewrite the sum of cubes as (x + 8) โx2 โ 8x + 64 โ . Analysis After writing the sum of cubes this way, we might think we should check to see if the trinomial portion can be factored further. However, the trinomial portion cannot be factored, so we do not need to check. 1.53 Factor the sum of cubes: 216a3 + b3. Example 1.54 Factoring a Difference of Cubes Factor 8x3 โ 125. Solution Notice that 8x3 and 125 are cubes because 8x3 = (2x)3 and 125 = 53. Write the difference of cubes as โ โ4x2 + 10x + 25 (2x โ 5) โ โ . Analysis Just as with the sum of cubes, we will not be able to further factor the trinomial portion. 1.54 Factor the difference of cubes: 1,000x3 โ 1. Factoring Expressions with Fractional or Negative Exponents Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance, 2x 1 4 + 5x 3 4 can be factored by pulling out x 1 4 and being rewritten as x 1 4 โ โ2 + 5x โ 1 2 โ โ. โ Example 1.55 86 Chapter 1 Prerequisites Factoring an Expression with Fractional or Negative Exponents Factor 3x(x + 2) โ1 3 + 4(x + 2) 2 3. Solution Factor out the term with the lowest value of the exponent. In this case, that would be (x + 2) โ 1 3. โ 1 3(3x + 4(x + 2)) โ 1 3(3x + 4x + 8) (x + 2) (x + 2) โ 1 3(7x + 8) (x + 2) Factor out the GCF. Simplify. 1.55 Factor 2(5a โ 1) 3 4 + 7a(5a โ 1) โ 1 4. Access these online resources for additional instruction and practice with fact
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oring polynomials. โข Identify GCF (http://openstaxcollege.org/l/findgcftofact) โข Factor Trinomials when a Equals 1 (http://openstaxcollege.org/l/facttrinom1) โข Factor Trinomials when a is not equal to 1 (http://openstaxcollege.org/l/facttrinom2) โข Factor Sum or Difference of Cubes (http://openstaxcollege.org/l/sumdifcube) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 87 1.5 EXERCISES Verbal 90v2 โ181v + 90 If the terms of a polynomial do not have a GCF, does 258. that mean it is not factorable? Explain. 278. 12t 2 + t โ 13 259. A polynomial is factorable, but it is not a perfect square trinomial or a difference of two squares. Can you factor the polynomial without finding the GCF? 260. How do you factor by grouping? Algebraic For the following exercises, find the greatest common factor. 261. 14x + 4xy โ 18xy2 262. 263. 264. 265. 266. 49mb2 โ 35m2 ba + 77ma2 30x3 y โ 45x2 y2 + 135xy3 200p3 m3 โ 30p2 m3 + 40m3 36 j4 k 2 โ 18 j3 k 3 + 54 j2 k 4 6y4 โ 2y3 + 3y2 โ y For the following exercises, factor by grouping. 267. 6x2 + 5x โ 4 268. 2a2 + 9a โ 18 269. 6c2 + 41c + 63 270. 271. 6n2 โ 19n โ 11 20w2 โ 47w + 24 272. 2p2 โ 5p โ 7 For the following exercises, factor the polynomial. 273. 7x2 + 48x โ 7 274. 10h2 โ 9h โ 9 275. 2b2 โ 25b โ 247 276. 9d 2 โ73d + 8 277. 279. 2n2 โ n โ 15 280. 16x2 โ 100 281. 25y2 โ 196 282. 121p2 โ 169 283. 4m2 โ 9 284. 361
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d 2 โ 81 285. 324x2 โ 121 286. 144b2 โ 25c2 287. 16a2 โ 8a + 1 288. 289. 290. 291. 292. 293. 49n2 + 168n + 144 121x2 โ 88x + 16 225y2 + 120y + 16 m2 โ 20m + 100 m2 โ 20m + 100 36q2 + 60q + 25 For the following exercises, factor the polynomials. 294. x3 + 216 295. 27y3 โ 8 296. 125a3 + 343 297. b3 โ 8d 3 298. 64x3 โ125 299. 729q3 + 1331 300. 125r 3 + 1,728s3 Chapter 1 Prerequisites Find the length of the base of the flagpole by 311. factoring. Extensions the following exercises, For completely. factor the polynomials 312. 16x4 โ 200x2 + 625 313. 81y4 โ 256 314. 16z4 โ 2,401a4 315. 316. 5x(3x + 2) โ 2 4 + (12x + 8) 3 2 โ โ โ32x3 + 48x2 โ 162x โ 243 โ โ1 88 301. 302. 303. 304. 305. 306. 307. 4x(x โ 1) โ 2 3 + 3(x โ 1) 1 3 3c(2c + 3) โ 1 4 โ 5(2c + 3) 3 4 3t(10t + 3) 1 3 + 7(10t + 3) 4 3 14x(x + 2) โ 2 5 + 5(x + 2) 3 5 9y(3y โ 13) 1 5 โ 2(3y โ 13) 6 5 5z(2z โ 9) โ 3 2 + 11(2z โ 9) 6d(2d + 3) โ 1 6 + 5(2d + 3) โ 1 2 5 6 Real-World Applications For the following exercises, consider this scenario: Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the cityโs parks. The park is a rectangle with an area of 98x2 + 105x โ 27 m2, as shown in the figure below. The length and width of the park are perfect factors of
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the area. Factor by grouping to find the length and width of the 308. park. A statue is to be placed in the center of the park. The 309. area of the base of the statue is 4x2 + 12x + 9m2. Factor the area to find the lengths of the sides of the statue. At the northwest corner of the park, the city is going 310. to install a fountain. The area of the base of the fountain is 9x2 โ 25m2. Factor the area to find the lengths of the sides of the fountain. For the following exercise, consider the following scenario: A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yd. as shown in the figure below. The flagpole will take up a square plot with area x2 โ 6x + 9 yd2. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 89 1.6 | Rational Expressions Learning Objectives In this section students will: 1.6.1 Simplify rational expressions. 1.6.2 Multiply rational expressions. 1.6.3 Divide rational expressions. 1.6.4 Add and subtract rational expressions. 1.6.5 Simplify complex rational expressions. A pastry shop has fixed costs of $280 per week and variable costs of $9 per box of pastries. The shopโs costs per week in terms of x, the number of boxes made, is 280 + 9x. We can divide the costs per week by the number of boxes made to determine the cost per box of pastries. 280 + 9x x Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions. Simplifying Rational Expressions The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Letโs start with the rational expression shown. We can factor the numerator and denominator to rewrite the expression. x2 + 8x + 16 x2 + 11x + 28 (x + 4)2 (x + 4)(x + 7) Then we
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can simplify that expression by canceling the common factor (x + 4). x + 4 x + 7 Given a rational expression, simplify it. 1. Factor the numerator and denominator. 2. Cancel any common factors. Example 1.56 Simplifying Rational Expressions Simplify x2 โ 9 x2 + 4x + 3. Solution 90 Chapter 1 Prerequisites (x + 3)(x โ 3) (x + 3)(x + 1) x โ 3 x + 1 Factor the numerator and the denominator. Cancel common factor (x + 3). Analysis We can cancel the common factor because any expression divided by itself is equal to 1. Can the x2 term be cancelled in Example 1.56? No. A factor is an expression that is multiplied by another expression. The x2 term is not a factor of the numerator or the denominator. 1.56 Simplify x โ 6 x2 โ 36. Multiplying Rational Expressions Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions. Given two rational expressions, multiply them. 1. Factor the numerator and denominator. 2. Multiply the numerators. 3. Multiply the denominators. 4. Simplify. Example 1.57 Multiplying Rational Expressions Multiply the rational expressions and show the product in simplest form: (x + 5)(x โ 1) 3(x + 6) โ
(2x โ 1) (x + 5) (x + 5)(x โ 1)(2x โ 1) 3(x + 6)(x + 5) (x + 5)(x โ 1)(2x โ 1) 3(x + 6)(x + 5) (x โ 1)(2x โ 1) 3(x + 6) Factor the numerator and denominator. Multiply numerators and denominators. Cancel common factors to simplify. Solution This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 91 (x + 5)(x โ 1) 3(x + 6) โ
(2x โ
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1) (x + 5) (x + 5)(x โ 1)(2x โ 1) 3(x + 6)(x + 5) (x + 5)(x โ 1)(2x โ 1) 3(x + 6)(x + 5) (x โ 1)(2x โ 1) 3(x + 6) Factor the numerator and denominator. Multiply numerators and denominators. Cancel common factors to simplify. 1.57 Multiply the rational expressions and show the product in simplest form: x2 + 11x + 30 x2 + 5x + 6 โ
x2 + 7x + 12 x2 + 8x + 16 Dividing Rational Expressions Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would x รท rewrite 1 x2 3 can multiply as we did before. as the product 1 x โ
3 x2. Once the division expression has been rewritten as a multiplication expression, we x โ
3 1 x2 = 3 x3 Given two rational expressions, divide them. 1. Rewrite as the first rational expression multiplied by the reciprocal of the second. 2. Factor the numerators and denominators. 3. Multiply the numerators. 4. Multiply the denominators. 5. Simplify. Example 1.58 Dividing Rational Expressions Divide the rational expressions and express the quotient in simplest form: Solution 2x2 + x โ 6 x2 โ 1 รท x2 โ 4 x2 + 2x + 1 9x2 โ 16 3x2 + 17x โ 28 รท 3x2 โ 2x โ 8 x2 + 5x โ 14 92 Chapter 1 Prerequisites 1.58 Divide the rational expressions and express the quotient in simplest form: 9x2 โ 16 3x2 + 17x โ 28 รท 3x2 โ 2x โ 8 x2 + 5x โ 14 Adding and Subtracting Rational Expressions Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Letโs look at an example of fraction addition. 5 24 + 1 40 + 3 120 = 25 120 = 28 120 = 7 30 We have to rewrite the fractions so they share a common denominator before we are
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able to add. We must do the same thing when adding or subtracting rational expressions. The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were (x + 3)(x + 4) and (x + 4)(x + 5), then the LCD would be (x + 3)(x + 4)(x + 5). Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of (x + 3)(x + 4) by x + 5 x + 5 and the expression with a denominator of (x + 4)(x + 5) by x + 3 x + 3. Given two rational expressions, add or subtract them. 1. Factor the numerator and denominator. 2. Find the LCD of the expressions. 3. Multiply the expressions by a form of 1 that changes the denominators to the LCD. 4. Add or subtract the numerators. 5. Simplify. Example 1.59 Adding Rational Expressions Add the rational expressions: 5 x + 6 y Solution First, we have to find the LCD. In this case, the LCD will be xy. We then multiply each expression by the appropriate form of 1 to obtain xy as the denominator for each fraction โ
5y xy + 6x xy This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 93 Now that the expressions have the same denominator, we simply add the numerators to find the sum. 6x + 5y xy Analysis Multiplying by y y or x x does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression. Example 1.60 Subtracting Rational Expressions Subtract the rational expressions: Solution 2 (x + 2)(x โ 2x + 2)2 โ 6 (x + 2)2 โ
6(x โ 2) (x + 2)2(x โ 2) 6x โ 12 โ (2x + 4) (x + 2)2(x โ
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2) โ 2 (x + 2)(x โ 2) 2(x + 2) (x + 2)2(x โ 2) 4x โ 16 (x + 2)2(x โ 2) 4(x โ 4) (x + 2)2(x โ 2) 6 x2 + 4x + 4 โ 2 x2 โ4 Factor. โ
x + 2 x + 2 Multiply each fraction to get LCD as denominator. Multiply. Apply distributive property. Subtract. Simplify. Do we have to use the LCD to add or subtract rational expressions? No. Any common denominator will work, but it is easiest to use the LCD. 1.59 Subtract the rational expressions: 3 x + 5 โ 1 xโ3. Simplifying Complex Rational Expressions A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression a can be simplified by rewriting the numerator 1 b + c 94 Chapter 1 Prerequisites as the fraction a 1 and combining the expressions in the denominator as 1 + bc b. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get a 1 โ
Given a complex rational expression, simplify it. b 1 + bc, which is equal to ab 1 + bc. 1. Combine the expressions in the numerator into a single rational expression by adding or subtracting. 2. Combine the expressions in the denominator into a single rational expression by adding or subtracting. 3. Rewrite as the numerator divided by the denominator. 4. Rewrite as multiplication. 5. Multiply. 6. Simplify. Example 1.61 Simplifying Complex Rational Expressions Simplify: y + 1 x x y. Solution Begin by combining the expressions in the numerator into one expression. x x + 1 x y โ
xy x + 1 x xy + 1 x Multiply by x x to get LCD as denominator. Add numerators. Now the numerator is a single rational expression and the denominator is a single rational expression. We can rewrite this as division, and then multiplication. xy + 1 x x y รท xy + 1 x x y xy + 1 y โ
x x y(xy +
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1) x2 Rewrite as multiplication. Multiply. 1.60 Simplify: y x x y โ y Can a complex rational expression always be simplified? Yes. We can always rewrite a complex rational expression as a simplified rational expression. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 95 Access these online resources for additional instruction and practice with rational expressions. โข Simplify Rational Expressions (http://openstaxcollege.org/l/simpratexpress) โข Multiply and Divide Rational Expressions (http://openstaxcollege.org/l/multdivratex) โข Add and Subtract Rational Expressions (http://openstaxcollege.org/l/addsubratex) โข Simplify a Complex Fraction (http://openstaxcollege.org/l/complexfract) 96 Chapter 1 Prerequisites 1.6 EXERCISES Verbal How can you use factoring to simplify rational 317. expressions? How do you use the LCD to combine two rational 318. expressions? 319. Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions. Algebraic For the expressions. following exercises, simplify the rational 320. 321. 322. 323. 324. 325. 326. 327. 328. 329. x2 โ 16 x2 โ 5x + 4 y2 + 10y + 25 y2 + 11y + 30 6a2 โ 24a + 24 6a2 โ 24 9b2 + 18b + 9 3b + 3 m โ 12 m2 โ 144 2x2 + 7x โ 4 4x2 + 2x โ 2 6x2 + 5x โ 4 3x2 + 19x + 20 a2 + 9a + 18 a2 + 3a โ 18 3c2 + 25c โ 18 3c2 โ 23c + 14 12n2 โ 29n โ 8 28n2 โ 5n โ 3 the For expressions and express the product in simplest form. following exercises, multiply the rational x2 โ x โ 6 2x2 + x โ 6 โ
2x2 + 7x โ 15 x2 โ 9 330. 331. This content is available for free at https://cnx.org/content/col11758/1.5 c2 + 2c โ 24 c2 +
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12c + 36 โ
c2 โ 10c + 24 c2 โ 8c + 16 332. 333. 334. 335. 336. 337. 338. 339. 2d 2 + 9d โ 35 d 2 + 10d + 21 โ
3d 2 + 2d โ 21 3d 2 + 14d โ 49 10h2 โ 9h โ 9 2h2 โ 19h + 24 โ
h2 โ 16h + 64 5h2 โ 37h โ 24 6b2 + 13b + 6 4b2 โ 9 โ
6b2 + 31b โ 30 18b2 โ 3b โ 10 2d 2 + 15d + 25 4d 2 โ 25 โ
2d 2 โ 15d + 25 25d 2 โ 1 6x2 โ 5x โ 50 15x2 โ 44x โ 20 โ
20x2 โ 7x โ 6 2x2 + 9x + 10 t 2 โ 1 t 2 + 4t + 3 โ
t 2 + 2t โ 15 t 2 โ 4t + 3 2n2 โ n โ 15 6n2 + 13n โ 5 โ
12n2 โ 13n + 3 4n2 โ 15n + 9 36x2 โ 25 6x2 + 65x + 50 โ
3x2 + 32x + 20 18x2 + 27x + 10 For the following exercises, divide the rational expressions. 340. 341. 342. 343. 344. 345. 346. 3y2 โ 7y โ 6 2y2 โ 3y โ 9 รท y2 + y โ 2 2y2 + y โ 3 6p2 + p โ 12 8p2 + 18p + 9 รท 6p2 โ 11p + 4 2p2 + 11p โ 6 q2 โ 9 q2 + 6q + 9 รท q2 โ 2q โ 3 q2 + 2q โ 3 18d 2 + 77d โ 18 27d 2 โ 15d + 2 รท 3d 2 + 29d โ 44 9d 2 โ 15d + 4 16x2 + 18x โ 55 32x2 โ 36x โ 11 รท 2x2 + 17x + 30 4x2 + 25x + 6 144b2 โ 25 72b2 โ 6b โ 10 รท 18b2 โ 21b + 5 36b2 โ 18b โ 10 16a
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2 โ 24a + 9 4a2 + 17a โ 15 รท 16a2 โ 9 4a2 + 11a + 6 Chapter 1 Prerequisites 97 347. 348. 22y2 + 59y + 10 12y2 + 28y โ 5 รท 11y2 + 46y + 8 24y2 โ 10y + 1 9x2 + 3x โ 20 3x2 โ 7x + 4 รท 6x2 + 4x โ 10 x2 โ 2x + 1 a b โ b a a + b ab 364. For the following exercises, add and subtract the rational expressions, and then simplify. 365. 3 + 4x 2x 2c c + 1 2c + 1 c + 1 349. x + 10 4 y 350. 351. 2q โ 6 12 3p 4 a + 1 + 5 a โ 3 352 353 354. x โ 1 x + 1 โ 2x + 3 2x + 1 355. 3z z + 1 + 2z + 5 z โ 2 356. 4p p + 1 โ p + 1 4p 357 the For expression. following exercises 2b 3a 358. 359. 360. 361. 362. 363. 366 Real-World Applications Brenda is placing tile on her bathroom floor. The area 367. of the floor is 15x2 โ 8x โ 7 ft2. The area of one tile is x2 โ 2x + 1ft2. To find the number of tiles needed, simplify the rational expression: 15x2 โ 8x โ 7 x2 โ 2x + 1. The area of Sandyโs yard is 25x2 โ 625 ft2. A patch 368. of sod has an area of x2 โ 10x + 25 ft2. Divide the two areas and simplify to find how many pieces of sod Sandy needs to cover her yard. Aaron wants to mulch his garden. His garden is 369. x2 + 18x + 81 ft2. One bag of mulch covers x2 โ 81 ft2. Divide the expressions and simplify to find how many bags of mulch Aaron needs to mulch his garden. Extensions For the following exercises, perform the given operations and simplify. x2 + x โ 6 x2 โ 2x โ 3 โ
2x2 โ 3x โ 9 x2 โ x โ 2 รท 10x2 + 27x + 18 x2 + 2x + 1 3y2 โ 10y + 3 3
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y2 + 5y โ 2 โ
2y2 โ 3y โ 20 2y2 โ y โ 15 y โ 4 370. 371. 372. simplify the rational 98 4a + 1 2a โ 3 + 2a โ 3 2a + 3 4a2 + 9 a 373. x2 + 7x + 12 x2 + x โ 6 รท 3x2 + 19x + 28 8x2 โ 4x โ 24 รท 2x2 + x โ 3 3x2 + 4x โ 7 Chapter 1 Prerequisites This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 99 CHAPTER 1 REVIEW KEY TERMS algebraic expression constants and variables combined using addition, subtraction, multiplication, and division associative property of addition the sum of three numbers may be grouped differently without affecting the result; in symbols, a + (b + c) = (a + b) + c associative property of multiplication the product of three numbers may be grouped differently without affecting the result; in symbols, a โ
(b โ
c) = (a โ
b) โ
c base in exponential notation, the expression that is being multiplied binomial a polynomial containing two terms coefficient any real number ai in a polynomial in the form an xn +... + a2 x2 + a1 x + a0 commutative property of addition two numbers may be added in either order without affecting the result; in symbols, a + b = b + a commutative property of multiplication two numbers may be multiplied in any order without affecting the result; in symbols, a โ
b = b โ
a constant a quantity that does not change value degree the highest power of the variable that occurs in a polynomial difference of squares opposite sign the binomial that results when a binomial is multiplied by a binomial with the same terms, but the distributive property the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, a โ
(b + c) = a โ
b + a โ
c equation a mathematical statement indicating that two expressions are equal exponent in exponential notation, the raised number or variable that indicates how many times the base is being multiplied exponential notation a shorthand method of writing products of the same factor factor by grouping a method for factoring a
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trinomial in the form ax2 + bx + c by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression formula an equation expressing a relationship between constant and variable quantities greatest common factor the largest polynomial that divides evenly into each polynomial identity property of addition there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, a + 0 = a identity property of multiplication there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols, a โ
1 = a index the number above the radical sign indicating the nth root integers the set consisting of the natural numbers, their opposites, and 0: { โฆ, โ3, โ2, โ1, 0, 1, 2, 3,โฆ} inverse property of addition there is a unique number, called the additive inverse (or opposite), denoted โ a, which, when added to the original number, results in the additive identity, 0; in symbols, a + (โa) = 0 for every real number a, 100 Chapter 1 Prerequisites inverse property of multiplication for every non-zero real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted 1 a, which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, a โ
1 a = 1 irrational numbers the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers leading coefficient the coefficient of the leading term leading term the term containing the highest degree least common denominator the smallest multiple that two denominators have in common monomial a polynomial containing one term natural numbers the set of counting numbers: {1, 2, 3,โฆ} order of operations operations a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to perfect square trinomial the trinomial that results when a binomial is squared polynomial a sum of terms each consisting of a variable raised to a nonnegative integer power principal nth root the number with the same sign as a that when raised to the nth power equals a principal square root the nonnegative square root of a number a that, when multiplied
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by itself, equals a radical the symbol used to indicate a root radical expression an expression containing a radical symbol radicand the number under the radical symbol rational expression the quotient of two polynomial expressions rational numbers the set of all numbers of the form m n, where m and n are integers and n โ 0. Any rational number may be written as a fraction or a terminating or repeating decimal. real number line a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left. real numbers the sets of rational numbers and irrational numbers taken together scientific notation a shorthand notation for writing very large or very small numbers in the form a ร 10 n where 1 โค |a| < 10 and n is an integer term of a polynomial any ai xi of a polynomial in the form an xn +... + a2 x2 + a1 x + a0 trinomial a polynomial containing three terms variable a quantity that may change value whole numbers the set consisting of 0 plus the natural numbers: {0, 1, 2, 3,โฆ} KEY EQUATIONS This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 101 Rules of Exponents For nonzero real numbers a and b and integers m and n Product rule Quotient rule Power rule Zero exponent rule Negative rule am โ
an = am + n am an = am โ n (am ) n = am โ
n a0 = 1 aโn = 1 an Power of a product rule (a โ
b) n = an โ
bn Power of a quotient rule n โ โ a b โ โ = an bn perfect square trinomial (x + a)2 = (x + a)(x + a) = x2 + 2ax + a2 difference of squares (a + b)(a โ b) = a2 โ b2 difference of squares a2 โ b2 = (a + b)(a โ b) perfect square trinomial a2 + 2ab + b2 = (a + b)2 sum of cubes โ โa2 โ ab + b2โ a3 + b3 = (a + b) โ difference of cubes โa2 + ab
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+ b2โ โ a3 โ b3 = (a โ b) โ KEY CONCEPTS 1.1 Real Numbers: Algebra Essentials โข Rational numbers may be written as fractions or terminating or repeating decimals. See Example 1.1 and Example 1.2. โข Determine whether a number is rational or irrational by writing it as a decimal. See Example 1.3. โข The rational numbers and irrational numbers make up the set of real numbers. See Example 1.4. A number can be classified as natural, whole, integer, rational, or irrational. See Example 1.5. 102 Chapter 1 Prerequisites โข The order of operations is used to evaluate expressions. See Example 1.6. โข The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. See Example 1.7. โข Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. See Example 1.8. They take on a numerical value when evaluated by replacing variables with constants. See Example 1.9, Example 1.10, and Example 1.12 โข Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. See Example 1.11 and Example 1.13. 1.2 Exponents and Scientific Notation โข Products of exponential expressions with the same base can be simplified by adding exponents. See Example 1.14. โข Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See Example 1.15. โข Powers of exponential expressions with the same base can be simplified by multiplying exponents. See Example 1.16. โข An expression with exponent zero is defined as 1. See Example 1.17. โข An expression with a negative exponent is defined as a reciprocal. See Example 1.18 and Example 1.19. โข The power of a product of factors is the same as the product of the powers of the same factors. See Example 1.20. โข The power of a quotient of factors is the same as the quotient of the powers of the same factors. See Example 1.21. โข The rules for exponential expressions can be combined to simplify more complicated expressions. See Example 1.22
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. โข Scientific notation uses powers of 10 to simplify very large or very small numbers. See Example 1.23 and Example 1.24. โข Scientific notation may be used to simplify calculations with very large or very small numbers. See Example 1.25 and Example 1.26. 1.3 Radicals and Rational Expressions โข The principal square root of a number a is the nonnegative number that when multiplied by itself equals a. See Example 1.27. โข โข If a and b are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b See Example 1.28 and Example 1.29. If a and b are nonnegative, the square root of the quotient a b is equal to the quotient of the square roots of a and b See Example 1.30 and Example 1.31. โข We can add and subtract radical expressions if they have the same radicand and the same index. See Example 1.32 and Example 1.33. โข Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See Example 1.34 and Example 1.35. โข The principal nth root of a is the number with the same sign as a that when raised to the nth power equals a. These roots have the same properties as square roots. See Example 1.36. โข Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See Example 1.37 and Example 1.38. โข The properties of exponents apply to rational exponents. See Example 1.39. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 103 1.4 Polynomials โข A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See Example 1.40. โข We can add and subtract polynomials by combining like terms. See Example 1.41 and Example 1.42. โข To multiply polynomials, use the distributive property to multiply each term in the first polynomial
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by each term in the second. Then add the products. See Example 1.43. โข FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See Example 1.44. โข Perfect square trinomials and difference of squares are special products. See Example 1.45 and Example 1.46. โข Follow the same rules to work with polynomials containing several variables. See Example 1.47. 1.5 Factoring Polynomials โข The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See Example 1.48. โข Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See Example 1.49. โข Trinomials can be factored using a process called factoring by grouping. See Example 1.50. โข Perfect square trinomials and the difference of squares are special products and can be factored using equations. See Example 1.51 and Example 1.52. โข The sum of cubes and the difference of cubes can be factored using equations. See Example 1.53 and Example 1.54. โข Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See Example 1.55. 1.6 Rational Expressions โข Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See Example 1.56. โข We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See Example 1.57. โข To divide rational expressions, multiply by the reciprocal of the second expression. See Example 1.58. โข Adding or subtracting rational expressions requires finding a common denominator. See Example 1.59 and Example 1.60. โข Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See Example 1.61. CHAPTER 1 REVIEW EXERCISES Real Numbers: Algebra Essentials 377. 5x + 9 = โ11 For the following exercises, perform the given operations. 374. (5 โ 3 โ
2)2 โ 6 375. 64 รท (2 โ
8) + 14 รท 7 376. 2 โ
52 + 6 รท 2 For the following exercises, solve the equation
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. 378. 2y + 42 = 64 For the following exercises, simplify the expression. 379. 9โ โy + 2โ โ รท 3 โ
2 + 1 380. 3m(4 + 7) โ m 104 Chapter 1 Prerequisites For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer. 381. 11 382. 0 383. 5 6 384. 11 Exponents and Scientific Notation For the following exercises, simplify the expression. 385. 22 โ
24 386. 45 43 387. โ a2 โ b3 4 โ โ 388. 6a2 โ
a0 2aโ4 389. (xy)4 y3 โ
2 x5 390. 4โ2 x3 yโ3 2x0 โ2 391. โ โ 2x2 y โ โ 396. 196 397. 361 398. 75 399. 162 400. 401. 32 25 80 81 402. 49 1250 403. 2 4 + 2 404. 4 3 + 6 3 405. 12 5 โ 13 5 406. 5 โ243 407. 3 250 3 โ8 Polynomials For the following exercises, perform the given operations and simplify. 408. โ โ โ3x3 + 2x โ 1 โ + โ โ โ4x2 โ 2x + 7 โ 392. โ 16a3 โ b2 โ โ4abโ1โ โ โ โ โ2 409. โ โ2y + 1โ โ โ โ โ โ2y2 โ 2y โ 5 โ the number in standard notation: 410. โ โ โ2x2 + 3x โ 6 โ + โ โ โ3x2 โ 4x + 9 โ 393. Write 2.1314 ร 10โ6 394. Write the number in scientific notation: 16,340,000 Radicals and Rational Expressions For the following exercises, find the principal square root. 395. 121 411. โ โ โ6a2 + 3
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a + 10 โ โ โ โ6a2 โ3a + 5 โ โ 412. (k + 3)(k โ 6) 413. (2h + 1)(3h โ 2) 414. โ โ โx2 + 1 (x + 1) โ This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 1 Prerequisites 105 Rational Expressions For the following exercises, simplify the expression. 434. x2 โ x โ 12 x2 โ 8x + 16 435. 4y2 โ 25 4y2 โ 20y + 25 436. 2a2 โ a โ 3 2a2 โ 6a โ 8 โ
5a2 โ 19a โ 4 10a2 โ 13a โ 3 437 โ 16 438. m2 + 5m + 6 2m2 โ 5m โ 3 รท 2m2 + 3m โ 9 4m2 โ 4m โ 3 439. 4d 2 โ 7d โ 2 6d 2 โ 17d + 10 รท 8d 2 + 6d + 1 6d 2 + 7d โ 10 440. 10 x + 6 y 441. 12 a2 + 2a + 1 โ 3 a2 โ1 442. d + 2 1 c 6c + 12d dc 443. 3 x โ 7 y 2 x 415. โ โ โm2 + 2m โ 3 (m โ 2) โ 416. (a + 2b)(3a โ b) 417. (x + y)(x โ y) Factoring Polynomials For the following exercises, find the greatest common factor. 418. 81p + 9pq โ 27p2 q2 419. 12x2 y + 4xy2 โ18xy 420. 88a3 b + 4a2 b โ 144a2 For the following exercises, factor the polynomial. 421. 2x2 โ 9x โ 18 422. 8a2 + 30a โ 27 423. d 2 โ 5d โ 66 424. x2 + 10x + 25 425. y2 โ 6y + 9 426. 4h2 โ 12hk + 9k 2 427. 361x2 โ 121 428. p3 + 216 429. 8x3 โ 125 430. 64q3 โ 27
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p3 431. 4x(x โ 1) โ 1 4 + 3(x โ 1) 3 4 432. 3pโ โp + 3โ โ 1 3 โ8โ โp + 3โ โ 4 3 433. 4r(2r โ 1) โ 2 3 โ 5(2r โ 1) 1 3 106 Chapter 1 Prerequisites CHAPTER 1 PRACTICE TEST For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer. 461. 3 4 โ8 625 444. โ13 445. 2 For the following exercises, evaluate the equations. 446. 2(x + 3) โ 12 = 18 447. y(3 + 3)2 โ 26 = 10 462. โ โ โ13q3 + 2q2 โ 3 โ โ โ โ โ6q2 + 5q โ 3 โ 463. โ โ โ6p2 + 2p + 1 โ + โ โ โ9p2 โ1 โ 464. โ โ โn2 โ 4n + 4 (n โ 2) โ 465. (a โ 2b)(2a + b) 448. Write the number in standard notation: 3.1415 ร 106 For the following exercises, factor the polynomial. 466. 16x2 โ 81 467. y2 + 12y + 36 468. 27c3 โ 1331 469. 3x(x โ 6) โ 1 4 + 2(x โ 6) 3 4 For the following exercises, simplify the expression. 470. 2z2 + 7z + 3 z2 โ 9 โ
4z2 โ 15z + 9 4z2 โ 1 471. x y + 2 x 472. a 2b โ 2b 9a 3a โ 2b 6a 449. Write 0.0000000212. the number in scientific notation: For the following exercises, simplify the expression. 450. โ2 โ
(2 + 3 โ
2)2 + 144 451. 4(x + 3) โ (6x + 2) 452. 35 โ
3โ3 453.
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