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Solution Manual for College Algebra, 7th Edition

Solution Manual for College Algebra, 7th Edition makes textbook problem-solving easy with a comprehensive guide that explains every step.

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    CONTENTS ¥ PROLOGUE: Principles of Problem Solving 1 CHAPTER P P REREQUISITES 3 P.1 Modeling the Real World with Algebra 3 P.2 Real Numbers 4 P.3 Integer Exponents and Scientific Notation 9 P.4 Rational Exponents and Radicals 14 P.5 Algebraic Expressions 18 P.6 Factoring 22 P.7 Rational Expressions 27 P.8 Solving Basic Equations 34 P.9 Modeling with Equations 39 Chapter P Review 45 Chapter P Test 51 ¥ FOCUS ON MODELING: Making the Best Decisions 54 CHAPTER 1 E QUATIONS AND GRAPHS 57 1.1 The Coordinate Plane 57 1.2 Graphs of Equations in Two Variables; Circles 65 1.3 Lines 79 1.4 Solving Quadratic Equations 90 1.5 Complex Numbers 98 1.6 Solving Other Types of Equations 101 1.7 Solving Inequalities 110 1.8 Solving Absolute Value Equations and Inequalities 129 1.9 Solving Equations and Inequalities Graphically 131 1.10 Modeling Variation 139 Chapter 1 Review 143 Chapter 1 Test 161 iii

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    iv Contents ¥ FOCUS ON MODELING: Fitting Lines to Data 165 CHAPTER 2 F UNCTIONS 169 2.1 Functions 169 2.2 Graphs of Functions 178 2.3 Getting Information from the Graph of a Function 190 2.4 Average Rate of Change of a Function 201 2.5 Linear Functions and Models 206 2.6 Transformations of Functions 212 2.7 Combining Functions 226 2.8 One-to-One Functions and Their Inverses 234 Chapter 2 Review 243 Chapter 2 Test 255 ¥ FOCUS ON MODELING: Modeling with Functions 259 CHAPTER 3 P OLYNOMIAL AND RATIONAL FUNCTIONS 267 3.1 Quadratic Functions and Models 267 3.2 Polynomial Functions and Their Graphs 276 3.3 Dividing Polynomials 291 3.4 Real Zeros of Polynomials 301 3.5 Complex Zeros and the Fundamental Theorem of Algebra 334 3.6 Rational Functions 344 Chapter 3 Review 377 Chapter 3 Test 395 ¥ FOCUS ON MODELING: Fitting Polynomial Curves to Data 398 CHAPTER 4 E XPONENTIAL AND LOGARITHMIC FUNCTIONS 401 4.1 Exponential Functions 401 4.2 The Natural Exponential Function 409 4.3 Logarithmic Functions 414 4.4 Laws of Logarithms 422 4.5 Exponential and Logarithmic Equations 426

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    Contents v 4.6 Modeling with Exponential Functions 433 4.7 Logarithmic Scales 438 Chapter 4 Review 440 Chapter 4 Test 448 ¥ FOCUS ON MODELING: Fitting Exponential and Power Curves to Data 450 CHAPTER 5 S YSTEMS OF EQUATIONS AND INEQUALITIES 455 5.1 Systems of Linear Equations in Two Variables 455 5.2 Systems of Linear Equations in Several Variables 462 5.3 Partial Fractions 470 5.4 Systems of Nonlinear Equations 481 5.5 Systems of Inequalities 488 Chapter 5 Review 500 Chapter 5 Test 508 ¥ FOCUS ON MODELING: Linear Programming 511 CHAPTER 6 M ATRICES AND DETERMINANTS 519 6.1 Matrices and Systems of Linear Equations 519 6.2 The Algebra of Matrices 530 6.3 Inverses of Matrices and Matrix Equations 538 6.4 Determinants and Cramer’s Rule 548 Chapter 6 Review 562 Chapter 6 Test 572 ¥ FOCUS ON MODELING: Computer Graphics 575 CHAPTER 7 C ONIC SECTIONS 579 7.1 Parabolas 579 7.2 Ellipses 584 7.3 Hyperbolas 593 7.4 Shifted Conics 600 Chapter 7 Review 612 Chapter 7 Test 622 ¥ FOCUS ON MODELING: Conics in Architecture 624

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    vi Contents CHAPTER 8 S EQUENCES AND SERIES 627 8.1 Sequences and Summation Notation 627 8.2 Arithmetic Sequences 632 8.3 Geometric Sequences 637 8.4 Mathematics of Finance 645 8.5 Mathematical Induction 649 8.6 The Binomial Theorem 658 Chapter 8 Review 662 Chapter 8 Test 669 ¥ FOCUS ON MODELING: Modeling with Recursive Sequences 670 CHAPTER 9 P ROBABILITY AND STATISTICS 673 9.1 Counting 673 9.2 Probability 680 9.3 Binomial Probability 688 9.4 Expected Value 693 Chapter 9 Review 695 Chapter 9 Test 701 ¥ FOCUS ON MODELING: The Monte Carlo Method 702 A PPENDIXES 705 A Calculations and Significant Figures 705 B Graphing with a Graphing Calculator 705

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    PROLOGUE: Principles of Problem Solving 1. Let r be the rate of the descent. We use the formula time  distance rate ; the ascent takes 1 15 h, the descent takes 1 r h, and the total trip should take 2 30  1 15 h. Thus we have 1 15  1 r  1 15  1 r  0, which is impossible. So the car cannot go fast enough to average 30 mi/h for the 2-mile trip. 2. Let us start with a given price P . After a discount of 40%, the price decreases to 0  6 P . After a discount of 20%, the price decreases to 0  8 P , and after another 20% discount, it becomes 0  8  0  8 P   0  64 P . Since 0  6 P  0  64 P , a 40% discount is better. 3. We continue the pattern. Three parallel cuts produce 10 pieces. Thus, each new cut produces an additional 3 pieces. Since the fi rst cut produces 4 pieces, we get the formula f  n   4  3  n  1  , n  1. Since f  142   4  3  141   427, we see that 142 parallel cuts produce 427 pieces. 4. By placing two amoebas into the vessel, we skip the fi rst simple division which took 3 minutes. Thus when we place two amoebas into the vessel, it will take 60  3  57 minutes for the vessel to be full of amoebas. 5. The statement is false. Here is one particular counterexample: Player A Player B First half 1 hit in 99 at-bats: average  1 99 0 hit in 1 at-bat: average  0 1 Second half 1 hit in 1 at-bat: average  1 1 98 hits in 99 at-bats: average  98 99 Entire season 2 hits in 100 at-bats: average  2 100 99 hits in 100 at-bats: average  99 100 6. Method 1: After the exchanges, the volume of liquid in the pitcher and in the cup is the same as it was to begin with. Thus, any coffee in the pitcher of cream must be replacing an equal amount of cream that has ended up in the coffee cup. Method 2: Alternatively, look at the drawing of the spoonful of coffee and cream mixture being returned to the pitcher of cream. Suppose it is possible to separate the cream and the coffee, as shown. Then you can see that the coffee going into the cream occupies the same volume as the cream that was left in the coffee. coffee cream Method 3 (an algebraic approach): Suppose the cup of coffee has y spoonfuls of coffee. When one spoonful of cream is added to the coffee cup, the resulting mixture has the following ratios: cream mixture  1 y  1 and coffee mixture  y y  1 . So, when we remove a spoonful of the mixture and put it into the pitcher of cream, we are really removing 1 y  1 of a spoonful of cream and y y  1 spoonful of coffee. Thus the amount of cream left in the mixture (cream in the coffee) is 1  1 y  1  y y  1 of a spoonful. This is the same as the amount of coffee we added to the cream. 7. Let r be the radius of the earth in feet. Then the circumference (length of the ribbon) is 2  r . When we increase the radius by 1 foot, the new radius is r  1, so the new circumference is 2   r  1  . Thus you need 2   r  1   2  r  2  extra feet of ribbon. 1

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    2 Principles of Problem Solving 8. The north pole is such a point. And there are others: Consider a point a 1 near the south pole such that the parallel passing through a 1 forms a circle C 1 with circumference exactly one mile. Any point P 1 exactly one mile north of the circle C 1 along a meridian is a point satisfying the conditions in the problem: starting at P 1 she walks one mile south to the point a 1 on the circle C 1 , then one mile east along C 1 returning to the point a 1 , then north for one mile to P 1 . That’s not all. If a point a 2 (or a 3 , a 4 , a 5 ,    ) is chosen near the south pole so that the parallel passing through it forms a circle C 2 ( C 3 , C 4 , C 5 ,    ) with a circumference of exactly 1 2 mile ( 1 3 mi, 1 4 mi, 1 5 mi,    ), then the point P 2 ( P 3 , P 4 , P 5 ,    ) one mile north of a 2 ( a 3 , a 4 , a 5 ,    ) along a meridian satis fi es the conditions of the problem: she walks one mile south from P 2 ( P 3 , P 4 , P 5 ,    ) arriving at a 2 ( a 3 , a 4 , a 5 ,    ) along the circle C 2 ( C 3 , C 4 , C 5 ,    ), walks east along the circle for one mile thus traversing the circle twice (three times, four times, fi ve times,    ) returning to a 2 ( a 3 , a 4 , a 5 ,    ), and then walks north one mile to P 2 ( P 3 , P 4 , P 5 ,    ).

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    P PREREQUISITES P.1 MODELING THE REAL WORLD WITH ALGEBRA 1. Using this model, we fi nd that if S  12, L  4 S  4  12   48. Thus, 12 sheep have 48 legs. 2. If each gallon of gas costs $3  50, then x gallons of gas costs $3  5 x . Thus, C  3  5 x . 3. If x  $120 and T  0  06 x , then T  0  06  120   7  2. The sales tax is $7  20. 4. If x  62,000 and T  0  005 x , then T  0  005  62,000   310. The wage tax is $310. 5. If   70, t  3  5, and d   t , then d  70  3  5  245. The car has traveled 245 miles. 6. V   r 2 h    3 2   5   45   141  4 in 3 7. (a) M  N G  240 8  30 miles/gallon (b) 25  175 G  G  175 25  7 gallons 8. (a) T  70  0  003 h  70  0  003  1500   65  5  F (b) 64  70  0  003 h  0  003 h  6  h  2000 ft 9. (a) V  9  5 S  9  5  4 km 3   38 km 3 (b) 19 km 3  9  5 S  S  2 km 3 10 . (a) P  0  06 s 3  0  06  12 3   103  7 hp (b) 7  5  0  06 s 3  s 3  125 so s  5 knots 11. (a) Depth (ft) Pressure (lb/in 2 ) 0 0  45  0   14  7  14  7 10 0  45  10   14  7  19  2 20 0  45  20   14  7  23  7 30 0  45  30   14  7  28  2 40 0  45  40   14  7  32  7 50 0  45  50   14  7  37  2 60 0  45  60   14  7  41  7 (b) We know that P  30 and we want to fi nd d , so we solve the equation 30  14  7  0  45 d  15  3  0  45 d  d  15  3 0  45  34  0. Thus, if the pressure is 30 lb/in 2 , the depth is 34 ft. 12. (a) Population Water use (gal) 0 0 1000 40  1000   40,000 2000 40  2000   80,000 3000 40  3000   120,000 4000 40  4000   160,000 5000 40  5000   200,000 (b) We solve the equation 40 x  120,000  x  120,000 40  3000. Thus, the population is about 3000. 13. The number N of cents in q quarters is N  25 q . 14. The average A of two numbers, a and b , is A  a  b 2 . 15. The cost C of purchasing x gallons of gas at $3  50 a gallon is C  3  5 x . 16. The amount T of a 15% tip on a restaurant bill of x dollars is T  0  15 x . 17. The distance d in miles that a car travels in t hours at 60 mi/h is d  60 t . 3

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    4 CHAPTER P Prerequisites 18. The speed r of a boat that travels d miles in 3 hours is r  d 3 . 19. (a) $12  3  $1   $12  $3  $15 (b) The cost C , in dollars, of a pizza with n toppings is C  12  n . (c) Using the model C  12  n with C  16, we get 16  12  n  n  4. So the pizza has four toppings. 20. (a) 3  30   280  0  10   90  28  $118 (b) The cost is  daily rental    days rented    cost per mile    miles driven  , so C  30 n  0  1 m . (c) We have C  140 and n  3. Substituting, we get 140  30  3   0  1 m  140  90  0  1 m  50  0  1 m  m  500. So the rental was driven 500 miles. 21. (a) (i) For an all-electric car, the energy cost of driving x miles is C e  0  04 x . (ii) For an average gasoline powered car, the energy cost of driving x miles is C g  0  12 x . (b) (i) The cost of driving 10,000 miles with an all-electric car is C e  0  04  10,000   $400. (ii) The cost of driving 10,000 miles with a gasoline powered car is C g  0  12  10,000   $1200. 22. (a) If the width is 20, then the length is 40, so the volume is 20  20  40  16,000 in 3 . (b) In terms of width, V  x  x  2 x  2 x 3 . 23. (a) The GPA is 4 a  3 b  2 c  1 d  0 f a  b  c  d  f  4 a  3 b  2 c  d a  b  c  d  f . (b) Using a  2  3  6, b  4, c  3  3  9, and d  f  0 in the formula from part (a), we fi nd the GPA to be 4  6  3  4  2  9 6  4  9  54 19  2  84. P.2 THE REAL NUMBERS 1. (a) The natural numbers are  1  2  3      . (b) The numbers       3   2   1  0  are integers but not natural numbers. (c) Any irreducible fraction p q with q   1 is rational but is not an integer. Examples: 3 2 ,  5 12 , 1729 23 . (d) Any number which cannot be expressed as a ratio p q of two integers is irrational. Examples are  2,  3,  , and e . 2. (a) ab  ba ; Commutative Property of Multiplication (b) a   b  c    a  b   c ; Associative Property of Addition (c) a  b  c   ab  ac ; Distributive Property 3. The set of numbers between but not including 2 and 7 can be written as (a)  x  2  x  7  in interval notation, or (b)  2  7  in interval notation. 4. The symbol  x  stands for the absolute value of the number x . If x is not 0, then the sign of  x  is always positive . 5. The distance between a and b on the real line is d  a  b    b  a  . So the distance between  5 and 2 is  2    5    7. 6. (a) Yes, the sum of two rational numbers is rational: a b  c d  ad  bc bd . (b) No, the sum of two irrational numbers can be irrational (     2  ) or rational (      0). 7. (a) No: a  b    b  a    b  a in general. (b) No; by the Distributive Property,  2  a  5    2 a   2   5    2 a  10    2 a  10. 8. (a) Yes, absolute values (such as the distance between two different numbers) are always positive. (b) Yes,  b  a    a  b  .

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    SECTION P.2 The Real Numbers 5 9. (a) Natural number: 100 (b) Integers: 0, 100,  8 (c) Rational numbers:  1  5, 0, 5 2 , 2  71, 3  14, 100,  8 (d) Irrational numbers:  7,   10. (a) Natural number:  16   4  (b) Integers:  500,  16,  20 5    4  (c) Rational numbers: 1  3, 1  3333    , 5  34,  500, 1 2 3 ,  16, 246 579 ,  20 5 (d) Irrational number:  5 11. Commutative Property of addition 12. Commutative Property of multiplication 13. Associative Property of addition 14. Distributive Property 15. Distributive Property 16. Distributive Property 17. Commutative Property of multiplication 18. Distributive Property 19. x  3  3  x 20. 7  3 x    7  3  x 21. 4  A  B   4 A  4 B 22. 5 x  5 y  5  x  y  23. 3  x  y   3 x  3 y 24.  a  b  8  8 a  8 b 25. 4  2 m    4  2  m  8 m 26. 4 3   6 y    4 3   6   y   8 y 27.  5 2  2 x  4 y    5 2  2 x   5 2  4 y    5 x  10 y 28.  3 a   b  c  2 d   3 ab  3 ac  6 ad 29. (a) 3 10  4 15  9 30  8 30  17 30 (b) 1 4  1 5  5 20  4 20  9 20 30. (a) 2 3  3 5  10 15  9 15  1 15 (b) 1  5 8  1 6  24 24  15 24  4 24  35 24 31. (a) 2 3  6  3 2   2 3  6  2 3  3 2  4  1  3 (b)  3  1 4  1  4 5    12 4  1 4  5 5  4 5   13 4  1 5  13 20 32. (a) 2 2 3  2 3 2  2  3 2  2 3  1 2  3  1 3  9 3  1 3  8 3 (b) 2 5  1 2 1 10  3 15  2 5  1 2 1 10  1 5  2 5  1 2 1 10  1 5  10 10  4  5 1  2  9 3  3 33. (a) 2  3  6 and 2  7 2  7, so 3  7 2 (b)  6   7 (c) 3  5  7 2 34. (a) 3  2 3  2 and 3  0  67  2  01, so 2 3  0  67 (b) 2 3   0  67 (c)  0  67    0  67  35. (a) False (b) True 36. (a) False:  3  1  73205  1  7325. (b) False 37. (a) True (b) False 38. (a) True (b) True 39. (a) x  0 (b) t  4 (c) a   (d)  5  x  1 3 (e)  p  3   5 40. (a) y  0 (b) z  1 (c) b  8 (d) 0    17 (e)  y     2 41. (a) A  B   1  2  3  4  5  6  7  8  (b) A  B   2  4  6  42. (a) B  C   2  4  6  7  8  9  10  (b) B  C   8 

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    6 CHAPTER P Prerequisites 43. (a) A  C   1  2  3  4  5  6  7  8  9  10  (b) A  C   7  44. (a) A  B  C   1  2  3  4  5  6  7  8  9  10  (b) A  B  C  ∅ 45. (a) B  C   x  x  5  (b) B  C   x   1  x  4  46. (a) A  C   x   1  x  5  (b) A  B   x   2  x  4  47.   3  0    x   3  x  0  _3 0 48.  2  8]   x  2  x  8  2 8 49. [2  8    x  2  x  8  2 8 50.   6   1 2    x   6  x   1 2  _6 1 _ 2 _ 51. [2      x  x  2  2 52.    1    x  x  1  1 53. x  1  x     1] 1 54. 1  x  2  x  [1  2] 1 2 55.  2  x  1  x    2  1] _2 1 56. x   5  x  [  5    _5 57. x   1  x    1    _1 58.  5  x  2  x    5  2  _5 2 59. (a) [  3  5] (b)   3  5] 60. (a) [0  2  (b)   2  0] 61.   2  0     1  1     2  1  _2 1 62.   2  0     1      1  0  _1 0 63. [  4  6]  [0  8   [0  6] 0 6 64. [  4  6]  [0  8   [  4  8  _4 8 65.     4    4    _4 4 66.    6]   2  10    2  6] 2 6

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    SECTION P.2 The Real Numbers 7 67. (a)  100   100 (b)  73   73 68. (a)     5  5        5  5   5   5, since 5   5. (b)  10     10   , since 10   . 69. (a)  6    4    6  4    2   2 (b)  1  1    1 1   1 70. (a)  2   12    2  12    10   10 (b)  1   1   1    1   1  1    1   0    1 71. (a)    2   6    12   12 (b)      1 3    15       5   5 72. (a)     6 24         1 4     1 4 (b)    7  12 12  7         5 5      1   1 73.    2   3    5   5 74.  2  5  1  5    4   4 75. (a)  17  2   15 (b)  21    3     21  3    24   24 (c)     3 10  11 8         12 40  55 40         67 40     67 40 76. (a)    7 15    1 21        49 105  5 105        54 105        18 35     18 35 (b)  38    57     38  57    19   19. (c)  2  6    1  8     2  6  1  8    0  8   0  8. 77. (a) Let x  0  777    . So 10 x  7  7777     x  0  7777     9 x  7. Thus, x  7 9 . (b) Let x  0  2888    . So 100 x  28  8888     10 x  2  8888     90 x  26. Thus, x  26 90  13 45 . (c) Let x  0  575757    . So 100 x  57  5757     x  0  5757     99 x  57. Thus, x  57 99  19 33 . 78. (a) Let x  5  2323    . So 100 x  523  2323     1 x  5  2323     99 x  518. Thus, x  518 99 . (b) Let x  1  3777    . So 100 x  137  7777     10 x  13  7777     90 x  124. Thus, x  124 90  62 45 . (c) Let x  2  13535    . So 1000 x  2135  3535     10 x  21  3535     990 x  2114. Thus, x  2114 990  1057 495 . 79.   3, so    3     3. 80.  2  1, so    1   2      2  1. 81. a  b , so  a  b     a  b   b  a . 82. a  b   a  b   a  b  b  a  2 b 83. (a)  a is negative because a is positive. (b) bc is positive because the product of two negative numbers is positive. (c) a  ba    b  is positive because it is the sum of two positive numbers. (d) ab  ac is negative: each summand is the product of a positive number and a negative number, and the sum of two negative numbers is negative. 84. (a)  b is positive because b is negative. (b) a  bc is positive because it is the sum of two positive numbers. (c) c  a  c    a  is negative because c and  a are both negative. (d) ab 2 is positive because both a and b 2 are positive. 85. Distributive Property

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    8 CHAPTER P Prerequisites 86. Day T O T G T O  T G  T O  T G  Sunday 68 77  9 9 Monday 72 75  3 3 Tuesday 74 74 0 0 Wednesday 80 75 5 5 Thursday 77 69 8 8 Friday 71 70 1 1 Saturday 70 71  1 1 T O  T G gives more information because it tells us which city had the higher temperature. 87. (a) When L  60, x  8, and y  6, we have L  2  x  y   60  2  8  6   60  28  88. Because 88  108 the post of fi ce will accept this package. When L  48, x  24, and y  24, we have L  2  x  y   48  2  24  24   48  96  144, and since 144   108, the post of fi ce will not accept this package. (b) If x  y  9, then L  2  9  9   108  L  36  108  L  72. So the length can be as long as 72 in.  6 ft. 88. Let x  m 1 n 1 and y  m 2 n 2 be rational numbers. Then x  y  m 1 n 1  m 2 n 2  m 1 n 2  m 2 n 1 n 1 n 2 , x  y  m 1 n 1  m 2 n 2  m 1 n 2  m 2 n 1 n 1 n 2 , and x  y  m 1 n 1  m 2 n 2  m 1 m 2 n 1 n 2 . This shows that the sum, difference, and product of two rational numbers are again rational numbers. However the product of two irrational numbers is not necessarily irrational; for example,  2   2  2, which is rational. Also, the sum of two irrational numbers is not necessarily irrational; for example,  2     2   0 which is rational. 89. 1 2   2 is irrational. If it were rational, then by Exercise 6(a), the sum  1 2   2     1 2    2 would be rational, but this is not the case. Similarly, 1 2   2 is irrational. (a) Following the hint, suppose that r  t  q , a rational number. Then by Exercise 6(a), the sum of the two rational numbers r  t and  r is rational. But  r  t     r   t , which we know to be irrational. This is a contradiction, and hence our original premise—that r  t is rational—was false. (b) r is rational, so r  a b for some integers a and b . Let us assume that rt  q , a rational number. Then by de fi nition, q  c d for some integers c and d . But then rt  q  a b t  c d , whence t  bc ad , implying that t is rational. Once again we have arrived at a contradiction, and we conclude that the product of a rational number and an irrational number is irrational. 90. x 1 2 10 100 1000 1 x 1 1 2 1 10 1 100 1 1000 As x gets large, the fraction 1  x gets small. Mathematically, we say that 1  x goes to zero. x 1 0  5 0  1 0  01 0  001 1 x 1 1 0  5  2 1 0  1  10 1 0  01  100 1 0  001  1000 As x gets small, the fraction 1  x gets large. Mathematically, we say that 1  x goes to in fi nity.

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    SECTION P.3 Integer Exponents and Scientific Notation 9 91. (a) Construct the number  2 on the number line by transferring the length of the hypotenuse of a right triangle with legs of length 1 and 1. 1 0 _1 2 1 Ï2 Ï 2 3 (b) Construct a right triangle with legs of length 1 and 2. By the Pythagorean Theorem, the length of the hypotenuse is  1 2  2 2   5. Then transfer the length of the hypotenuse to the number line. 1 0 _1 2 1 Ï5 3 Ï5 (c) Construct a right triangle with legs of length  2 and 2 [construct  2 as in part (a)]. By the Pythagorean Theorem, the length of the hypotenuse is    2  2  2 2   6. Then transfer the length of the hypotenuse to the number line. 1 0 _1 2 1 Ï2 Ï 2 Ï 6 Ï6 Ï2 3 92. (a) Subtraction is not commutative. For example, 5  1   1  5. (b) Division is not commutative. For example, 5  1   1  5. (c) Putting on your socks and putting on your shoes are not commutative. If you put on your socks fi rst, then your shoes, the result is not the same as if you proceed the other way around. (d) Putting on your hat and putting on your coat are commutative. They can be done in either order, with the same result. (e) Washing laundry and drying it are not commutative. (f) Answers will vary. (g) Answers will vary. 93. Answers will vary. 94. (a) If x  2 and y  3, then  x  y    2  3    5   5 and  x    y    2    3   5. If x   2 and y   3, then  x  y    5   5 and  x    y   5. If x   2 and y  3, then  x  y    2  3   1 and  x    y   5. In each case,  x  y    x    y  and the Triangle Inequality is satis fi ed. (b) Case 0: If either x or y is 0, the result is equality, trivially. Case 1: If x and y have the same sign, then  x  y      x  y if x and y are positive   x  y  if x and y are negative      x    y  . Case 2: If x and y have opposite signs, then suppose without loss of generality that x  0 and y  0. Then  x  y    x  y    x    y  . P.3 INTEGER EXPONENTS AND SCIENTIFIC NOTATION 1. Using exponential notation we can write the product 5  5  5  5  5  5 as 5 6 . 2. Yes, there is a difference:   5  4    5    5    5    5   625, while  5 4    5  5  5  5    625. 3. In the expression 3 4 , the number 3 is called the base and the number 4 is called the exponent . 4. When we multiply two powers with the same base, we add the exponents. So 3 4  3 5  3 9 . 5. When we divide two powers with the same base, we subtract the exponents. So 3 5 3 2  3 3 . 6. When we raise a power to a new power, we multiply the exponents. So  3 4  2  3 8 .

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    10 CHAPTER P Prerequisites 7. (a) 2  1  1 2 (b) 2  3  1 8 (c)  1 2   1  2 (d) 1 2  3  2 3  8 8. Scientists express very large or very small numbers using scienti fi c notation. In scienti fi c notation, 8,300,000 is 8  3  10 6 and 0  0000327 is 3  27  10  5 . 9. (a) No,  2 3   2   3 2  2  9 4 . (b) Yes,   5  4  625 and  5 4    5 4    625. 10. (a) No,  x 2  3  x 2  3  x 6 . (b) No,  2 x 4  3  2 3   x 4  3  8 x 12 . 11. (a)  2 6   64 (b)   2  6  64 (c)  1 5  2    3  3  1 2   3  3 5 2   27 25 12. (a)   5  3   125 (b)  5 3   125 (c)   5  2   2 5  2    5  2  2  2 5 2  4 13. (a)  5 3  0  2  1  1 2 (b) 2  3 3 0  1 2 3  1 8 (c)  1 4   2  4 2  16 14. (a)  2  3    2  0   1 2 3   1 8 (b)  2 3    2  0   2 3   8 (c)   2 3   3  3 3   2  3   27 8 15. (a) 5 3  5  5 4  625 (b) 3 2  3 0  3 2  9 (c)  2 2  3  2 6  64 16. (a) 3 8  3 5  3 13  1,594,323 (b) 6 0  6  6 (c)  5 4  2  5 8  390,625 17. (a) 5 4  5  2  5 2  25 (b) 10 7 10 4  10 3  1000 (c) 3 2 3 4  1 3 2  1 9 18. (a) 3  3  3  1  3  4  1 3 4  1 81 (b) 5 4 5  5 3  125 (c) 7 2 7 5  1 7 3  1 343 19. (a) x 2 x 3  x 2  3  x 5 (b)   x 2  3    1  3 x 2  3   x 6 (c) t  3 t 5  t  3  5  t 2 20. (a) y 5  y 2  y 5  2  y 7 (b)  8 x  2  8 2 x 2  64 x 2 (c) x 4 x  3  x 4  3  x 21. (a) x  5  x 3  x  5  3  x  2  1 x 2 (b)   2   4  5    2  4  5    1  1  (c) y 10 y 0 y 7  y 10  0  7  y 3 22. (a) y 2  y  5  y 2  5  y  3  1 y 3 (b) z 5 z  3 z  4  z 5  3  4  z  2  1 z 2 (c) x 6 x 10  x 6  10  1 x 4 23. (a) a 9 a  2 a  a 9  2  1  a 6 (b)  a 2 a 4  3   a 2  4  3   a 6  3  a 6  3  a 18 (c)  2 x  2  5 x 6   2 2 x 2  5 x 6  20 x 2  6  20 x 8 24. (a) z 2 z 4 z 3 z  1  z 2  4 z 3  1  z 6 z 2  z 6  2  z 4 (b)  2 a 3 a 2  4   2 a 3  2  4   2 a 5  4  2 4 a 5  4  16 a 20 (c)   3 z 2  3  2 z 3     3  3 z 2  3  2 z 3   54 z 6  3   54 z 9 25. (a)  3 x 2 y   2 x 3   3  2 x 2  3 y  6 x 5 y (b)  2 a 2 b  1   3 a  2 b 2   2  3 a 2  2 b  1  2  6 b

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    SECTION P.3 Integer Exponents and Scientific Notation 11 (c)  4 y 2   x 4 y  2  4 y 2 x 4  2 y 2  4 x 8 y 2  2  4 x 8 y 4 26. (a)  4 x 3 y 2   7 y 5   4  7 x 3 y 2  5  28 x 3 y 7 (b)  9 y  2 z 2   3 y 3 z   9  3 y  2  3 z 2  1  27 yz 3 (c)  8 x 7 y 2   1 2 x 3 y   2  8 x 7 y 2  1 2 x 3 y  2  2 2  8 x 7 y 2 x 3  2 y 2  32 x 7 y 2 x 6 y 2  32 x 7  6 y 2  2  32 x 27. (a)  2 x 2 y 3  2  3 y   2 2 x 2  2 y 3  2  3 y  12 x 4 y 7 (b) x 2 y  1 x  5  x 2    5  y  1  x 7 y  1  x 7 y (c)  x 2 y 3  3  x 2  3 y 3 3 3  x 6 y 3 27 28. (a)  5 x  4 y 3   8 x 3  2  5 x  4 y 3  8 2 x 3  2  5  8 2 x  4  6 y 3  320 x 2 y 3 (b) y  2 z  3 y  1  y y 2 z 3  1 yz 3 (c)  a 3 b  2 b 3  2  a 6 b  4 b 6  a 6 b 10 29. (a)  x 3 y 3   1  1 x 3 y 3 (b)  a 2 b  2   3  a 3   2  a 2   3  b  2   3  a 3   2   a  6 b 6 a  6  b 6 a 12 (c)  x 2 y  2   2  2 y  3 x 2  3  x 2   2  y    2   2   2 3 y  3  3  x  2  3   x  4 y  4  8 y  9 x  6  8 x  4  6 y  4  9  8 x 10 y 13 30. (a)  x  2 y 4   3   y 4 x 2   3   x 2 y 4  3  x 6 y 12 (b)  y 2   1  2 x  3 y 4   3  y  2 2  3 x  3   3  y 4   3   x 9 8 y 14 (c)  2 a  1 b  2   3  b  1 2 a 2  2  2  3 a  1   3  b    2   3  b  1  2  2  2 a  2  2   1 32 ab 8 31. (a) 3 x  2 y 5 9 x  3 y 2  x y 3 3 (b)  2 x 3 y  1 y 2   2   2 x 3 y 3   2  y 3  2 2 2 x 3  2  y 6 4 x 6 (c)  y  1 x  2   1  3 x  3 y 2   2  y  1   1  x    2   1  3  2 x  3   2  y  2   2   x 4 y 5 9 32. (a) 1 2 a  3 b  4 2 a  5 b  1  1 2  2  1 a  3    5  b  4    1   1 4 a 2 b  3  a 2 4 b 3

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