ANSWERS A-13 82. a) ⎛ − ⎝⎜ ⎞ ⎠⎟ = ⋅ − = ⋅ ⋅ = 4 4 4 4 12, 4 4 4 4 15, 4 4 4 4 16, ⋅ + = ⎛ + ⎝⎜ ⎞ ⎠⎟ = 4 4 4 4 17, 4 4 4 4 20 b) − = 44 4 4 10 SECTION 5.4, PAGE 256 69. Between 6 and ≈ 6.5; 37 6.08. 70. Between 7.5 and ≈ 8; 61 7.81. 71. Between 9.5 and ≈ 10; 97 9.85. 72. Between 11 and ≈ 11.5; 123 11.09. 73. Between 13 and ≈ 13.5; 170 13.04. 74. Between 14 and ≈ 14.5; 200 14.14. 79. False. c may be a rational number or an irrational number for a composite number c. For example, 25 is a rational number; 8 is an irrational number. 83. False. The product of a rational number and an irrational number may be a rational number or an irrational number. The product of 0 and an irrational number is 0, which is a rational number. The product of a nonzero rational number and an irrational number is always an irrational number. 95. b) Irrational = … 0.7 0.8366600265 . Since the decimal number is not a terminating or a repeating decimal number, this number is an irrational number. SECTION 5.5, PAGE 263 Answer to Recreational Math box on page 260 163 123 71 22 4 24 24 2 4 1 3 1 3 4 3 2 1 1 4 2 2 3 4 29. Commutative property of addition. The only difference between the expressions on both sides of the equal sign is the order of 5 and x. 30. Associative property of addition. On the left side of the equal sign, the x and 5 are in parentheses. On the right side of the equal sign, the 5 and 6 are in parentheses. 66. 2 2 3 1 4 4 1 2 3 1 4 3 2 3 2 4 1 22 24 243 32 71 51 12 SECTION 5.7, PAGE 280 45. = ⋅ ⎛ ⎝⎜ ⎞ ⎠⎟ − a 2 1 2 n n 1 47. = − ⋅ ⎛ ⎝⎜ ⎞ ⎠⎟ − a 16 1 2 n n 1 48. = − − − a 3( 2) n n 1 74. c) It is risky because it is likely that you won’t have enough money to continue doubling your previous bet if you lose several times in a row. Also, there may be a maximum amount you can bet in a game of chance. SECTION 5.8, PAGE 287 18. Each number in the Fibonacci sequence is either a prime number or is relatively prime with the number preceding or succeeding it in sequence. Therefore, the GCF of any two consecutive Fibonacci numbers is 1. 30. = = = ≈ = = 1 1 1, 2 1 2, 3 2 1.5, 5 3 1.667, 8 5 1.6, 13 8 1.625, ≈ ≈ ≈ ≈ 21 13 1.615, 34 21 1.619, 55 34 1.6176, 89 55 1.6182. The consecutive ratios alternate, increasing and decreasing about the golden ratio. Chapter 6 SECTION 6.1, PAGE 304 81. − = ( 1) 1 n for any even number n, since an even number of factors of −( 1) when multiplied will always be 1. SECTION 6.5, PAGE 336 9. 4 10. 25 24 23 22 21 0 4 5 1 2 3 11. 25 24 23 22 21 0 4 5 1 2 3 12. 25 24 23 22 21 0 4 5 1 2 3 13. 25 14. 21029 28 27 26 25 21 0 23 24 22 15. 25 24 23 22 21 0 4 5 1 2 3 16. 21 17. 26 0 18. 23 0 19. 29 20. 8 21. 4 0 22. 25 24 23 22 21 0 4 5 1 2 3 23. 22 21 0 1 2 3 7 8 4 5 6 24. 28 27 26 25 24 23 22 21 0 1 2 25. 21 0123456789 26. 21 0 27. 25 24 23 22 21 0 4 5 1 2 3 28. 25 24 23 22 21 0 4 5 1 2 3 29. No solution 0 30. 25 24 23 22 21 0 No solution 4 5 1 2 3
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