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Solution Manual for College Algebra and Trigonometry, 4th Edition

Solution Manual for College Algebra and Trigonometry, 4th Edition offers the best solutions to textbook problems, helping you prepare for exams and assignments.

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    S OLUTION M ANUAL B EVERLY F USFIELD C OLLEGE A LGEBRA AND T RIGONOMETRY AND P RECALCULUS A R IGHT T RIANGLE A PPROACH F OURTH E DITION J. S. Ratti University of South Florida Marcus McWaters University of South Florida Leslaw A. Skrzypek University of South Florida

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    CONTENTS Chapter P Basic Concepts of Algebra P.1 The Real Numbers and Their Properties ...................................................... 1 P.2 Integer Exponents and Scientific Notation ................................................... 7 P.3 Polynomials ................................................................................................ 13 P.4 Factoring Polynomials................................................................................ 20 P.5 Rational Expressions .................................................................................. 27 P.6 Rational Exponents and Radicals ............................................................... 37 Chapter P Review Exercises.................................................................................... 46 Chapter P Practice Test ........................................................................................... 51 Chapter 1 Equations and Inequalities 1.1 Linear Equations in One Variable .............................................................. 53 1.2 Applications of Linear Equations: Modeling ............................................. 63 1.3 Quadratic Equations ................................................................................... 74 1.4 Complex Numbers: Quadratic Equations with Complex Solutions ........... 89 1.5 Solving Other Types of Equations ............................................................. 96 1.6 Inequalities ............................................................................................... 116 1.7 Equations and Inequalities Involving Absolute Value ............................. 131 Chapter 1 Review Exercises .................................................................................. 148 Chapter 1 Practice Test A ...................................................................................... 159 Chapter 1 Practice Test B ...................................................................................... 160 Chapter 2 Graphs and Functions 2.1 The Coordinate Plane ............................................................................... 163 2.2 Graphs of Equations ................................................................................. 172 2.3 Lines ......................................................................................................... 185 2.4 Functions .................................................................................................. 200 2.5 Properties of Functions............................................................................. 210 2.6 A Library of Functions ............................................................................. 219 2.7 Transformations of Functions .................................................................. 230 2.8 Combining Functions; Composite Functions ........................................... 248 2.9 Inverse Functions ..................................................................................... 263 Chapter 2 Review Exercises .................................................................................. 275 Chapter 2 Practice Test A ...................................................................................... 285 Chapter 2 Practice Test B ...................................................................................... 286 Cumulative Review Exercises (Chapters P − 2)...................................................... 287 Chapter 3 Polynomial and Rational Functions 3.1 Quadratic Functions ................................................................................. 291 3.2 Polynomial Functions ............................................................................... 310 3.3 Dividing Polynomials ............................................................................... 323 3.4 The Real Zeros of a Polynomial Function ................................................ 334 3.5 The Complex Zeros of a Polynomial Function ........................................ 353 3.6 Rational Functions.................................................................................... 363 3.7 Variation ................................................................................................... 382 Chapter 3 Review Exercises .................................................................................. 387 Chapter 3 Practice Test A ...................................................................................... 403 Chapter 3 Practice Test B ...................................................................................... 404 Cumulative Review Exercises (Chapters P − 3)...................................................... 405

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    Chapter 4 Exponential and Logarithmic Functions 4.1 Exponential Functions .............................................................................. 408 4.2 Logarithmic Functions ............................................................................. 420 4.3 Rules of Logarithms ................................................................................. 433 4.4 Exponential and Logarithmic Equations and Inequalities ........................ 446 4.5 Logarithmic Scales; Modeling ................................................................. 459 Chapter 4 Review Exercises .................................................................................. 470 Chapter 4 Practice Test A ...................................................................................... 478 Chapter 4 Practice Test B ...................................................................................... 479 Cumulative Review Exercises (Chapters P − 4)...................................................... 480 Chapter 5 Trigonometric Functions 5.1 Angles and Their Measure ....................................................................... 483 5.2 Right-Triangle Trigonometry ................................................................... 490 5.3 Trigonometric Functions of Any Angle; The Unit Circle ........................ 502 5.4 Graphs of the Sine and Cosine Functions................................................. 514 5.5 Graphs of the Other Trigonometric Functions ......................................... 526 5.6 Inverse Trigonometric Functions ............................................................. 538 Chapter 5 Review Exercises .................................................................................. 546 Chapter 5 Practice Test A ...................................................................................... 551 Chapter 5 Practice Test B ...................................................................................... 552 Cumulative Review Exercises (Chapters P − 5)...................................................... 552 Chapter 6 Trigonometric Identities and Equations 6.1 Verifying Identities .................................................................................. 555 6.2 Sum and Difference Formulas .................................................................. 568 6.3 Double-Angle and Half-Angle Formulas ................................................. 582 6.4 Product-to-Sum and Sum-to-Product Formulas ....................................... 598 6.5 Trigonometric Equations I........................................................................ 606 6.6 Trigonometric Equations II ...................................................................... 616 Chapter 6 Review Exercises .................................................................................. 637 Chapter 6 Practice Test A ...................................................................................... 644 Chapter 6 Practice Test B ...................................................................................... 645 Cumulative Review Exercises (Chapters 1 − 6) ...................................................... 647 Chapter 7 Applications of Trigonometric Functions 7.1 The Law of Sines...................................................................................... 650 7.2 The Law of Cosines.................................................................................. 664 7.3 Areas of Polygons Using Trigonometry ................................................... 678 7.4 Vectors ..................................................................................................... 693 7.5 The Dot Product ....................................................................................... 703 7.6 Polar Coordinates ..................................................................................... 713 7.7 Polar Form of Complex Numbers; DeMoivre’s Theorem ....................... 729 Chapter 7 Review Exercises .................................................................................. 744 Chapter 7 Practice Test A ...................................................................................... 752 Chapter 7 Practice Test B ...................................................................................... 754 Cumulative Review Exercises (Chapters 1–7) ...................................................... 756

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    Chapter 8 Systems of Equations and Inequalities 8.1 Systems of Linear Equations in Two Variables ....................................... 759 8.2 Systems of Linear Equations in Three Variables ..................................... 777 8.3 Partial-Fraction Decomposition ............................................................... 798 8.4 Systems of Nonlinear Equations .............................................................. 821 8.5 Systems of Inequalities............................................................................. 835 8.6 Linear Programming................................................................................. 848 Chapter 8 Review Exercises .................................................................................. 863 Chapter 8 Practice Test A ...................................................................................... 876 Chapter 8 Practice Test B ...................................................................................... 879 Cumulative Review Exercises (Chapters P − 8)...................................................... 881 Chapter 9 Matrices and Determinants 9.1 Matrices and Systems of Equations.......................................................... 884 9.2 Matrix Algebra ......................................................................................... 905 9.3 The Matrix Inverse ................................................................................... 925 9.4 Determinants and Cramer’s Rule ............................................................. 946 Chapter 9 Review Exercises .................................................................................. 959 Chapter 9 Practice Test A ...................................................................................... 970 Chapter 9 Practice Test B ...................................................................................... 971 Cumulative Review Exercises (Chapters P − 9)...................................................... 972 Chapter 10 The Conic Sections 10.2 The Parabola ............................................................................................. 975 10.3 The Ellipse................................................................................................ 990 10.4 The Hyperbola ........................................................................................ 1008 Chapter 10 Review Exercises .............................................................................. 1032 Chapter 10 Practice Test A .................................................................................. 1041 Chapter 10 Practice Test B .................................................................................. 1043 Cumulative Review Exercises (Chapters P − 10).................................................. 1044 Chapter 11 Further Topics in Algebra 11.1 Sequences and Series ............................................................................. 1047 11.2 Arithmetic Sequences; Partial Sums ...................................................... 1058 11.3 Geometric Sequences and Series ............................................................ 1066 11.4 Mathematical Induction .......................................................................... 1075 11.5 The Binomial Theorem .......................................................................... 1086 11.6 Counting Principles ................................................................................ 1094 11.7 Probability .............................................................................................. 1101 Chapter 11 Review Exercises .............................................................................. 1105 Chapter 11 Practice Test A .................................................................................. 1109 Chapter 11 Practice Test B .................................................................................. 1109 Cumulative Review Exercises (Chapters P − 11).................................................. 1110

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    1 Chapter P Basic Concepts of Algebra P.1 The Real Numbers and Their Properties P.1 Practice Problems 1. Let x = 2.132132132…. Then, 1000 x = 2132.132132… 1000 2132.132132 2.132132 999 2130 2130 710 999 333 x x x x        2. a. Natural numbers: 2, 7 b. Whole numbers: 0, 2, 7 c. Integers: 21 6, 3, 0, 2, 7 7     d. Rational numbers: 21 1 4 6, 3, , 0, , 2, 7 7 2 3      e. Irrational numbers: 3, 17 f. Real numbers: the set B 3. a. 3 2 8  b.      2 2 3 3 3 9   a a a a c. 4 1 1 1 1 1 1 2 2 2 2 2 16            4. a. True b. True c. False 5.     3, 1, 0, 1, 3 , 4, 2, 0, 2, 4       A B     0 , 4, 3, 2, 1, 0, 1, 2, 3, 4       A B A B   6. a.   1 2 , I I     b.   1 2 2, 5 I I    7. a. 10 10   b. 3 4 1 1     c.   2 3 7 1 1     8.   7, 2 7 2 9 9        d 9. a.   3 5 20 15 20 5        b. 5 12 6 2 5 2 2 5 4 1          c. 9 1 8 5 7 5 7 2 35 33 4 4           d.   2 3 4 x    for x = 6   2 2 3 6 4 3 2 3 4 1           10. a. 7 3 14 3 17 4 8 8 8 8     b. 8 2 40 6 34 3 5 15 15 15     c. 9 7 9 14 3   3 14 2 7  1 3 1 3 2  d. 5 5 15 5 16 5 8 15 8 16 8 15 16      1 8 1 16  2 15 3 2 3  11. a.     2 3 3 2 3 3 1 10 1 3 3 3 3 3 x x              b. 1 1 22 7 7 7 3 3 3 x y        12. 39 C 4 13 4 17 C 3        P.1 Concepts and Vocabulary 1. Whole numbers are formed by adding the number zero to the set of natural numbers. 2. The number − 3 is an integer, but it is also a rational number and a real number. 3. If a < b , then a is to the left of b on the number line. 4. If a real number is not a rational number, it is an irrational number.

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    2 Chapter P Basic Concepts of Algebra 5. True 6. False. 5 1 2 2 2    7. True 8. False. An example is     2 2 2 2 4,     which is rational. P.1 Building Skills 9. 0.3 , repeating 10. 0.6 , repeating 11. 0.8  , terminating 12. 0.25  , terminating 13. 0.27 , repeating 14. 0.3 , repeating 15. 3.16 , repeating 16. 2.73 , repeating 17. 375 15 3.75 100 4   18. 235 47 2.35 100 20      19. 10 53.3 5.3 9 48 48 16 9 3 16 5.3 3 x x x x         20. 10 96.6 9.6 9 87 87 29 9 3 x x x x      21. 100 213.13 2.13 99 211 211 99 x x x x     22. 100 323.23 3.23 99 320 320 99 x x x x     23. 100 452.323 4.523 99 447.8 447.8 4478 2239 99 990 495 x x x x       24. 100 142.3535 1.4235 99 140.93 140.93 14, 093 99 9900 x x x x      25. Rational 26. Rational 27. Rational 28. Rational 29. Rational 30. Rational 31. Irrational 32. Irrational Exercises 33-38 refer to the set 12 17 A 19, , 3, 0, 2, 10, , 11 3 4           33. Natural numbers: 2, 11 34. Whole numbers: 0, 2, 11 35. Integers: 12 19, 4, 0, 2, 11 3     36. Rational numbers: 12 17 19, , 0, 2, , 11 3 4   37. Irrational numbers: 3, 10  38. Real numbers: All numbers in set A are real numbers. 39. 3 10 base: 10; exponent: 3 3 10 10 10 10 1000     40. 4 5 base: 5; exponent: 4 4 5 5 5 5 5 625      41. 3 2 3       base: 2 ; 3 exponent: 3 3 2 2 2 2 8 3 3 3 3 27           42. 4 5 2       base: 5 ; 2 exponent: 4 4 5 5 5 5 5 625 2 2 2 2 2 16           

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    Section P.1 The Real Numbers and Their Properties 3 43.   3 2  base: –2; exponent: 3         3 2 2 2 2 8          44. 4 1 2        base: 1 ; 2  exponent: 4 4 1 1 1 1 1 1 2 2 2 2 2 16                                      45. 4 2 3  base: 3; exponent: 4 4 2 3 2 3 3 3 3 2 81 162          46. 3 1 5 3        base: 1 ; 3 exponent: 3 3 1 1 1 1 1 5 5 5 5 3 3 3 3 27 27               47. 4 2 3   base: 3; exponent: 4 4 2 3 2 3 3 3 3 2 81 162              48.   4 3 2    base: –2; exponent: 4        4 3 2 3 2 2 2 2 3 16 48               49.   5 3 2    base: –2; exponent: 5           5 3 2 3 2 2 2 2 2 3 32 96                50.   2 5 3    base: –3; exponent: 2      2 5 3 5 3 3 5 9 45             51. 3 2   52. 3 2    53. 1 1 2 2  54. 1   x x 55. 5 2  x 56. 1 2   x 57. 0 x   58. 0 x  59. 2 7 14   x 60. 2 3 5   x 61. 24 4 6  62. 5 2    63. 4 0   64. 9 1 4 2 2    65.   4, 3, 2, 0, 1, 2, 3, 4     A B  66.   0, 2, 4  A B  67.   4, 2, 0, 2 A C     68.   4, 3, 2, 1, 0, 1, 2, 3, 4 B C       69.       3, 0, 2 4, 3, 2, 0, 2, 4 B C A A          70.       4, 3, 2, 1, 0, 2, 4 3, 0, 2, 4 A C B B           71.       4, 3, 2, 0, 1, 2, 3, 4 4, 3, 2, 0, 2 A B C C            72.       4, 3, 2, 0, 1, 2, 3, 4 4, 3, 2, 1, 0, 1, 2, 3, 4 A B C C             73.     1 2 1 2 2, 5 ; 1, 3 I I I I      74.     1 2 1 2 1, 7 ; 3, 5 I I I I     75.   1 2 1 2 6, 10 ; I I I I       76.   1 2 1 2 , ; I I I I        77.     1 2 1 2 , ; 2, 5 I I I I       78.     1 2 1 2 2, ; 0, I I I I        79.       1 2 1 2 , ; 1, 3 5, 7 I I I I         80.       1 2 1 2 , 2 6, ; 3, 0 I I I I         81. 4 4   82. 17 17     83. 5 5 7 7   84. 3 3 5 5   85. 5 2 5 2    86. 2 5 5 2    87. 3 2 2 3    88. 3 3      89. 8 8 1 8 8    90. 8 8 1 8 8     

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    4 Chapter P Basic Concepts of Algebra 91. 5 7 5 7 2 2        92. 4 7 4 7 3 3       93. (3,8) 3 8 5 5      d 94. (2,14) 2 14 12 12      d 95. ( 6, 9) 6 9 15 15        d 96. ( 12, 3) 12 3 15 15        d 97. ( 20, 6) 20 ( 6) 14 14          d 98. ( 14, 1) 14 ( 1) 13 13          d 99. 22 4 22 4 26 26 , 7 7 7 7 7 7                   d 100. 16 3 16 3 19 19 , 5 5 5 5 5 5                   d 101. 3 1    x 102. 6 2     x 103. 3   x 104. 0  x 105. 5  x 106. 1   x 107. 3 9 4 4    x 108. 1 3 2     x 109. 4( 1) 4 4    x x 110. ( 3)(2 ) 6 3      x x 111. 5( 1) 5 5 5      x y x y 112. 2(3 5 ) 6 10 2      x y x y 113. Additive inverse: 5  ; reciprocal: 1 5 114. Additive inverse: 2 3 ; reciprocal: 3 2  115. Additive inverse: 0; no reciprocal 116. Additive inverse: 1.7  ; reciprocal: 10 17 117. Additive inverse 118. Additive inverse 119. Multiplicative identity 120. Multiplicative identity 121. Associative property of multiplication

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    Section P.1 The Real Numbers and Their Properties 5 122. Associative property of multiplication 123. Multiplicative inverse 124. Multiplicative inverse 125. Additive identity 126. Additive identity 127. Associative property of addition 128. Commutative and associative properties of addition 129. 3 4 9 20 29 5 3 15 15 15     130. 7 3 14 15 29 10 4 20 20 20     131. 6 5 42 25 67 5 7 35 35 35     132. 9 5 54 5 59 2 12 12 12 12     133. 5 3 25 9 34 17 6 10 30 30 30 15      134. 8 2 24 10 34 15 9 45 45 45     135. 5 9 25 36 11 8 10 40 40 40      136. 7 1 35 8 27 8 5 40 40 40     137. 5 7 55 63 8 9 11 99 99 99      138. 5 7 15 14 1 8 12 24 24 24     139. 2 1 4 5 1 5 2 10 10 10      140. 1 1 3 2 1 4 6 12 12 12     141. 3 8 2 4 27 9   142. 9 14 2 7 27 3   143. 8 8 16 8 15 3 5 16 5 15 5 16 2 15      144. 5 5 15 5 6 1 6 15 6 6 6 15 3 6      145. 7 7 21 7 16 2 8 21 8 16 8 21 3 16      146. 3 3 7 3 15 9 10 7 10 15 10 7 14 15      147. 3 1 3 1 2 5 1 10 2 2 2 2       148. 7 3 14 3 11 2 2 2 2 2 2      149. 2 1 2 1 6 5 1 3 15 3 5 3 15 15 15        150. 5 3 10 3 20 9 11 2 3 2 3 2 6 6 6        151. 2( ) 3 2(3 ( 5)) 3( 5) 2( 2) ( 15) 4 15 11                x y y 152. 2( ) 5 2(3 ( 5)) 5( 5) 2( 2) ( 25) 4 ( 25) 21                    x y y 153. 3 2 3 3 2 5 3(3) 2(5) 9 10 1           x y 154. 7 7 3 ( 5) 7 8 7(8) 56        x y 155. 3 3 3( 5) 3( 5) 2 2 3 ( 15) ( 15) 2 18 ( 15) 9 ( 15) 6 2                     x y xy 156. 3 5 3 3( 5) 3 2 43 ( 15) 3 3             y xy x 157. 2(1 2 ) 2(1 2(3)) ( ) ( 3)( 5) 5 2( 5) 15 2 15 13 5 x x y y                 

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    6 Chapter P Basic Concepts of Algebra 158. 3(2 ) 3(2 3) (1 ) (1 3( 5)) 5 3( 1) (1 ( 15)) 5 3 77 16 5 5                    x xy y 159. 14 1 14 1 31 4 62 2 3 2 ( 5) 6 5 15 4 4          x y 160. 4 8 4 8 4 8 2 ( 5) 3 5 5 3 5 2 2 52 2 104 15 5 75                                   y x y 161. 3 3 3 x x x xy y y    162.    2 2 2 3 3 2 6 5 6 x x x x x x x          163.     5 3 5 5 3 5 15 x x x      . 164.    2 25 4 100 x x x  165.   3 2 3 2 x y x y      166.   2 4 5 2 4 5 x y x y      167. 1 x y x y y x x x x      168. x y x y x z x z x z       169.     1 1 1 x y xy x y       170. 1 1 1 1 1 1 y y x x y x x y      171.     2 , 2 2 2 2 , 2 2 2 a b a b a a b d P M a a b b a b b a d Q M b                  Since     , , , 2 2 a b b a d P M d Q M       M is the midpoint of the line segment PQ . 172. Answers may vary. Using the hint, we have 4 40 13 130    and 7 70 . 13 130  Therefore, 40 31 30 0 130 130 130 1 10 69 70 . 130 130 130 130                 P.1 Applying the Concepts 173. a. people who own either MP3 players or people who own DVD players. b. people who own both MP3 players and DVD players. 174. a. A = {Lexus LS 460} b. B = {Lexus LS 460, Lincoln Continental} c. C = {Lexus LS 460, Lincoln Continental, Audi A4} d. A B  = {Lexus LS 460} e. B C  = {Lexus LS 460, Lincoln Continental } f. A B  = {Lexus LS 460, Lincoln Continental } g. A C  = {Lexus LS 460, Lincoln Continental, Audi A4} 175. 119.5 134.5   x 176. 30 107   x 177. a. 124 120 4   b. 137 120 17   c. 114 120 6 6     178. a. 15 14 1   b. 15 17.5 2.5   c. 15 9 6   179. Let x = the number of calories from broccoli. Then we have 522.5 55 0 522.5 55 9.5       x x x The number of grams of broccoli is 9.5 × 100 = 950 grams.

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    Section P.2 Integer Exponents and Scientific Notation 7 180. Let x = the number of orders of french fries. The number of calories lost from broccoli is 6 × 55 = 330. Then we have 165 330 0 165 330 2       x x x So, Carmen will have to eat 2 orders of french fries. P.1 Beyond the Basics 181. True 182. True 183. a. False. For example, 0 2 0.   b. The products of a nonzero rational number and an irrational number is an irrational number. 184. False. For example, 2 2 2.   185. False. For example,     2 3 6 3 4.      186. False. For example, 12 4 2. 3   187. True 188. True For exercises 189–190, use the following definition: An integer P is even if p = 2 n for some integer n . An integer q is odd if q = 2 k + 1 for some integer k . 189. a. If a is odd, then a = 2 m + 1 for some integer m .            2 2 2 2 1 2 1 2 1 4 4 1 4 1 1 2 2 1 1, a m m m m m m m m m               which is of the format 2 k +1. Therefore, 2 a is an odd integer. b. Assume that b is odd. Then, from part (a), 2 b is also odd. However, 2 b is an even integer, so, by contradiction, b must be an even integer. 190. 2 2 q is even because it is of the form 2 n , so it follows that 2 p is even. Then, from exercise 189(b), p is also even. Therefore, p = 2 n , for some integer n . Substituting, we have   2 2 2 2 2 2 2 2 2 2 q p n q n       2 2 2 , q n  and, thus, q is even using exercise 189 (b). P.1 Getting Ready for the Next Section 191. a. 5 2 3 a a a   b. 4 7 11 a a a   c. m n m n a a a    192. a. 3 2 b b b  b. 7 4 3 b b b  c. m m n n b b b   193. a.   3 6 2 a a  b.   2 8 4 a a  c.   n m mn a a  194. a.   2 2 2 ab a b  b.   4 4 4 ab a b  c.   n n n ab a b  P.2 Integer Exponents and Scientific Notation P.2 Practice Exercises 1. a. 1 1 2 2   b. 0 4 1 5        c. 2 2 3 2 4 2 3 9                2. a. 2 7 9 3 3   x x x b.       3 3 2 3 3 3 3 0 2 4 4 4 16 16 16 1 16 x x x x x x            3. a. 4 4 0 4 0 3 3 3 81 3    

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    8 Chapter P Basic Concepts of Algebra b. 1 ( 2) 3 2 5 5 5 125 5       c. 3 3 ( 4) 7 4 2 2 2 3 3 3      x x x x 4. a.   0 5 ( 5)(0) 0 7 7 7 1      b.   5 0 (0)( 5) 0 7 7 7 1      c.   8 1 ( 1)(8) 8 8 1 x x x x       d.   5 2 ( 2)( 5) 10       x x x 5. a.   1 1 1 1 1 1 2 2 2 2                         x x x x b.     2 2 1 2 1 2 ( 1)(2) 2 2 5 5 5 25 25         x x x x x c.   3 2 3 2(3) 3 6   x y x y x y d.   6 3 2 ( 2)( 3) 3 3        x x y x y y 6. a. 2 2 2 1 1 1 3 9 3         b. 2 2 2 2 10 7 7 49 7 10 100 10                 7. a.   2 2 4 4 8 1 1 2 2 4          x x x b.     3 2 2 3 3 3 6 3 2 1      x y x y x y xy xy 8. 5 732, 000 7.32 10   9. 10 2 2 8 2.9 10 290 10 0.89 10 $89 325 3.25 10        per person P.2 Concepts and Vocabulary 1. In the expression 2 7 , the number 2 is called the exponent. 2. In the expression 7 3 ,  the base is 3. 3. The number 2 1 4  simplifies to be the positive integer 16. 4. The power-of-a-product rule allows us to rewrite   3 5 a as 3 3 5 . a 5. False.   10 10 11 11   6. False. When   3 2 x is simplified, the expression becomes 6 . x 7. True 8. False P.2 Building Skills 9. 2 2 1 1 1 1 3 3 3 3 9                       10. 3 3 1 1 1 1 1 2 2 2 2 2 8                             11. 4 4 1 2 16 2          12.   2 2 1 2 4 2            13. 0 7 1  14.   0 8 1   15.   0 7 1     16.   0 2 1  17.   2 3 3 2 6 2 2 2 64     18.   3 2 2 3 6 3 3 3 729     19.     2 2 2 2 4 4 1 1 3 3 3 81 3        20.     1 2 1 2 2 2 1 1 7 7 7 49 7       

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    Section P.2 Integer Exponents and Scientific Notation 9 21.   3 3 3 2 2 1 1 1 5 25 15, 625 5                 22.   3 3 3 1 1 1 1 5 5 5 125                 23.     3 5 ( 3 5) 2 4 4 4 4 16        24.     2 3 ( 2 3) 1 7 7 7 7 7        25.   0 0 3 10 1 1 2     26.   0 0 5 9 1 1 0     27. 2 2 2 2 1 1 1 1 1 2 3 3 3 3 9 9 9                          28. 2 2 2 2 1 1 1 1 1 2 5 5 5 5 25 25 25                          29. 3 3 1 1 2 8 2       30. 2 2 1 1 3 9 3       31.     2 2 1 1 3 9 3      32.     3 3 1 1 2 8 2       33. 11 (11 10) 1 10 2 2 2 2 2     34. 6 (6 8) 2 8 2 3 1 1 3 3 9 3 3       35.   4 3 12 12 12 5 5 1 5 5   36.   2 5 10 (10 8) 2 8 8 9 9 9 9 81 9 9      37. 5 2 (5 4) ( 2 ( 3)) 1 1 4 3 2 3 2 3 2 3 6 2 3              38. 2 3 ( 2 ( 3)) (3 1) 3 1 2 4 5 4 5 4 5 4 5 4 25 100                39. 2 1 1 2 5 1 1 2 2 2 25 25 2 5            40. 2 1 1 2 7 1 1 3 3 3 49 49 3 7            41. 1 2 3 3 2         42. 1 1 5 5         43. 2 2 2 2 2 3 3 9 3 2 4 2                 44. 2 2 2 2 3 2 2 4 2 3 9 3                 45. 2 2 11 1 1 49 121 7 121 11 49 7                 46. 2 2 13 1 1 25 169 5 169 13 25 5                 47. 4 0 4 4 1    x y x x 48. 1 0 1 1 1     x y x x 49. 1 1     y x y y x x 50. 2 2 2 2 2 2 1     x x y x y y 51. 1 2 2 1    x y xy 52. 3 2 3 2 1    x y x y 53.   4 3 12 12 1     x x x 54.   2 5 10 10 1     x x x 55.   3 11 ( 11) ( 3) 33 x x x        56.   12 4 ( 4) ( 12) 48 x x x       

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    10 Chapter P Basic Concepts of Algebra 57. 5 5 5 3( ) 3    x y x y 58. 6 6 6 8( ) 8    x y x y 59.   2 2 1 2 2 2 4 4 4 x xy x y y     60.   3 3 1 3 3 3 6 6 6 y x y x y x     61.   5 5 1 ( 1) ( 5) 5 5 3 3 3 x x y x y y         62.   6 6 1 6 ( 1) ( 6) 6 5 5 5 y xy x y x            63.     2 3 3 2 6 6 10 4 5 2 5 10 4 2 1 x x x x x x x x x          64.   2 2 2 2 12 10 4 3 4 12 10 3 1 x x x x x x x x x         65. 3 3 3 3 3 3 3 6 3 2 2 3 6 3 3 3 3 2 2 8 8 8 8               xy x y x y x y x x x y x y x 66. 4 4 4 4 4 4 4 12 4 3 3 4 12 4 8 4 8 5 5 625 625 625 625               xy x y x y x y x x x y x y x 67. 5 2 5 2 5 5 10 5 5 5 10 5 5 5 5 3 ( 3) 243 243 243                  x y x y x y x x x x y x y 68. 3 2 3 3 2 3 3 6 3 3 3 6 3 3 3 2 ( 2) 8 8 8                  x y x y x y y y y x y x y 69. 2 2 2 2 2 2 2 1 1 1 3 ( 3) 25 9 9 1 1 5 5 9 25 25 x x x x x                  70. 4 4 4 4 4 4 4 4 4 1 1 1 5 ( 5) 5 625 1 1 3 3 81 3 81 625                  y y y y y 71. 3 2 3 2 3 6 5 3 5 3 3 15 9 15 4 4 64 64 x x x xy x y x y x y               72. 5 2 5 2 5 5 10 5 10 3 3 5 15 10 3 3 243 243            x y x y x y x y y y y 73. 3 3 5 3 ( 2) 3 1 5 4 2 4           x y x x y x y x y y 74. 2 2 3 2 ( 1) 2 2 3 4 1 2 4           x y x x y x y x y y 75. 3 5 3 ( 4) 5 7 1 2 4 7 2 27 3 3 3 9           x y x x y x y x y y 76. 5 2 5 7 2 ( 3) 2 1 7 3 2 15 5 5 5 3           x y y x y x y x y x 77.       3 2 4 3 2 3 4 3 6 4 3 3 6 4 1 1 1 x x x x x x x x x x x x                78.         12 12 12 4 3 4 3 4 4 12 4 3 12 4 12 12 12 3 4 12 4 12 12 12 12 12 12 12 4 12 12 12 12 12 4 12 0 0 16 16 16 2 2 8 8 2 2 2 2 2 2 2 1 2 a a a b a b b b a b a b a b a b a a b b a b a b b b                                                

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    Section P.2 Integer Exponents and Scientific Notation 11 79.                      3 2 3 4 2 2 2 4 2 2 3 3 2 4 4 4 2 2 3 2 3 3 6 2 4 2 4 8 8 3 8 6 8 5 2 5 2 2 2 2 16 16 1 16 1 1 16 16 xy xy x y x y x y x y x y x y x y x y x y x y x y                            80.                  2 2 3 2 4 2 3 3 4 5 5 5 2 6 1 5 5 2 6 5 1 5 2 2 2 6 6 12 6 2 2 2 12 2 x y y x y y x y xy x y x y x y xy x y y x y x y x                                                           81.                 2 3 2 2 7 3 2 3 2 7 3 2 2 2 2 3 2 2 7 7 3 2 7 4 4 4 x y z x y z x y z x y z x y z x y z    3 2 2 2 2 6 4 2 3 7 2 7 7 21 14 7 6 21 4 14 2 7 15 10 5 15 10 5 16 16 16 16 16 x y z x y z x y z x y z x y z x y z x y z                82.         2 2 2 2 2 2 2 2 2 3 2 2 1 1 6 2 1 1 1 3 2 2 2 2 1 2 5 2 2 5 2 1 3 3 0 3 3 2 2 4 4 4 4 4 xyz x y z x y z x y x z x y x z x y xz x y z x y z x y z z x y z x                   83. 2 2 2 4 1 ( 3) 2 2 4 3 2 4 6 4 6 5 5 5 5 1             a bc a b c a b c b a b a 84. 2 5 2 5 2 2 5 2 2 3 2 4 3 2 3 4 2 3 4 3 3 1 2 2 ( 3) ( ) 9 9 8 ( 2) 8 9 9 8 8                a bc a b c a b c a b c a b c a a b c bc 85. 3 3 2 3 ( 3)( 3) ( 2)( 3) 2 4 3 2( 3) ( 4)( 3) 3( 3) 3 9 6 6 12 9 3 ( 6) 9 12 6 ( 9) 3 15 3 3 15 3                                   xy z x y z x y z x y z x y z x y z x y z x z x y z y 86. 1 2 1 1 ( 2)( 1) ( 1)( 1) 5 8 ( 5)( 1) 1 ( 8)( 1) 1 2 1 1 5 2 ( 1) 1 8 5 1 8 3 6 3 7 6 7 xy z x y z x yz x y z x y z x y z x y z y x y z x z                                    87. 2 125 1.25 10   88. 2 247 2.47 10   89. 5 850, 000 8.5 10   90. 5 205, 000 2.05 10   91. 3 0.007 7.0 10    92. 3 0.0019 1.9 10    93. 6 0.00000275 2.75 10    94. 6 0.0000038 3.8 10    P.2 Applying the Concepts 95. 3 3 3 135 ft 2 1080 ft   96. 3 3 3 1 675 in. 25 in. 3         97. a. 2 2 2 (2 ) 4 4 2    x x A A b. 2 2 2 (3 ) 9 9 3    x x A A

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