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01_Chapter R - Basic Algebraic Operations

# 01_Chapter R - Basic Algebraic Operations - 1 CHAPTER R...

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1 CHAPTER R Section R-1 1. True 3. False 5. False 7. True 9. False 11. True 13. 1 15. Division by zero is not defined. Undefined. 17. 0 ÷ (9·8) = 0 ÷ 72 = 0 19. 0 · 100 + 1 100 ± ² ³ ´ µ = 0 since a ·0 = 0· a = 0 for all real numbers a . 21. 4 9 + 12 5 = 4 ± 5 + 12 ± 9 9 ± 5 = 20 + 108 45 = 128 45 23. - 1 100 + 4 25 ± ² ³ ´ µ = - 1 100 + 16 100 ± ² ³ ´ µ = - 17 100 or ± 17 100 25. 3 8 ± ² ³ ´ µ · 1 + 2 -1 = 8 3 + 1 2 = 8 ± 2 + 1 ± 3 3 ± 2 = 16 + 3 6 = 19 6 27. Commutative (·) 29. Distributive 31. Inverse (·) 33. Inverse (+) 35. Identity (+) 37. Negatives (Theorem 1) 39. The even integers between -3 and 5 are -2, 0, 2, and 4. Hence, the set is written {-2, 0, 2, 4}. 41. The letters in "status" are: s,t,a,u . Hence, the set is written { s,t,a,u }, or, equivalently, { a,s,t,u }. 43. Since there are no months starting with B , the set is empty. ± 45. (A) The empty set, ± , is a subset of every set. { a } and { b } are subsets of S 2 . S 2 is a subset of itself. Thus, there are four subsets of S 2 . (B) The empty set, ± , is a subset of every set. { a }, { b }, and { c } are one-member subsets of S 3 . { b, c }, { a, c }, and { a, b } are two-member subsets of S 3 . S 3 is a subset of itself. Thus, there are 1 + 3 + 3 + 1 = 8 subsets of S 3 . (C) The empty set, ± , is a subset of every set. { a }, { b }, { c }, and { d } are one-member subsets of S 4 . { a, b }, { a, c }, { a, d }, { b, c }, { b, d }, and { c, d } are two-member subsets of S 4 . { b, c, d }, { a, c, d }, { a, b, d }, and { b, c, d } are three-member subsets of S 4 . S 4 is a subset of itself. Thus, there are 1 + 4 + 6 + 4 + 1 = 16 subsets of S 4 . 47. Yes. This restates Zero Property (2). (Theorem 2, Part 2) 49. (A) True. (B) False, 2 3 is an example of a real number that is not irrational. (C) True.

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2 CHAPTER R BASIC ALGEBRAIC OPERATIONS 51. 3 5 and -1.43 are two examples of infinitely many. 53. (A) {1, 144 } (B) {-3, 0, 1, 144 } (C) ± 3, ± 2 3 ,0,1, 9 5 , 144 ² ³ ´ µ · (D) { 3 } 55. (A) 0.888 888…; repeating; repeated digit: 8 (B) 0.272 727…; repeating; repeated digits: 27 (C) 2.236 067 977…; nonrepeating and nonterminating (D) 1.375; terminating 57. (A) True; commutative property for addition. (B) False; for example 3 - 5 ± 5 - 3. (C) True; commutative property for multiplication. (D) False; for example 9 ÷ 3 ± 3 ÷ 9. 59. F 3 – 8 = -5 is one of many counterexamples. 61. T 63. F 2 · 2 2 = 1 is one of many counterexamples. 65. T 67. (A) List each element of A . Follow these with each element of B that is not yet listed. {1,2,3,4,6} (B) List each element of A that is also an element of B {2,4}. 69. Let c = 0.090909… Then 100 c = 9.0909… 100 c - c = (9.0909…) - (0.090909…) 99 c = 9 c = 9 99 = 1 11 71. 23 23 · 12 = 23(2 + 10) 12 = 23 · 2 + 23 · 10 46 23 · 2 230 23 · 10 = 46 + 230 276 = 276 Section R-2 1. 256 3. 2 3 ± ² ³ ´ µ 4 = 2 4 3 4 = 16 81 5. 4 -4 = 1 4 4 = 1 256 7. 625 9. (-3) -1 = 1 ± 3 = ± 1 3 11. -7 -2 = - 1 7 2 = ± 1 49 13. 1 15. 10 17. 5 19. 9 -3/2 = 1 9 32 = 1 9 12 ( ) 3 = 1 3 3 = 1 27 21. Irrational 23. 1 -3 + 3 -1 = 1 1 3 + 1 3 1 = 1 1 + 1 3 = 3 + 1 3 = 4 3 25. 5 5 4 = 1 5 3 = 1 125 27. x 5 x -2 = x 5+(-2) = x 3 29. (2 y )(3 y 2 )(5 y 4 ) = 2·3·5 yy 2 y 4 = 30 y 1+2+4 = 30 y 7 31. ( a 2 b 3 ) 5 = ( a 2 ) 5 ( b 3 ) 5 = a 10 b 15 33. u 1/3 u 5/3 = u (1/3)+(5/3) = u 6/3 = u 2 35. ( x -3 ) 1/6 = x -3/6 = x -1/2 = 1 x 12 37. (49 a 4 b -2 ) 1/2 = 49 1/2 ( a 4 ) 1/2 ( b -2 ) 1/2 = 7 a 2 b -1 = 7 a 2 b
SECTION R-2 3 39. 45,320,000 = 4.5320000. ± 10 7 7 places left = 4.532 ± 10 7 41. 0.066 = 0.06.6 ± 10 -2 = 6.6 ± 10 -2 2 places right negative exponent 43. 0.000 000 084 = 8.4 ± 10 -8 8 places right 45. 9 ± 10 -5 = 0.00009. = 0.000 09 5 places left 47. 3.48 ± 10 6 = 3.480 000. = 3,480,000 6 places right 49. 4.2 ± 10 -9 = 0.000 000 004.2 = 0.000 000 004 2 9 places left 51.

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