264 CHAPTER 5 Number Theory and the Real Number System 36. Give an example to show that the associative property of multiplication may be true for the negative integers. (2)(3)( 4) (2) (3)( 4) 24 [ ] − − − =−⎡⎣− − ⎤⎦=− 37. Does the associative property hold for the integers under the operation of division? Give an example to support your answer. No. ÷ ÷ ≠ ÷ ÷ (16 8) 2 16 (8 2). 38. Does the associative property hold for the integers under the operation of subtraction? Give an example to support your answer. No. − − ≠ − − (10 4) 2 10 (4 2). 39. Does the associative property hold for the real numbers under the operation of division? Give an example to support your answer. No. ÷ ÷ ≠ ÷ ÷ (81 9) 3 81 (9 3). 40. Does + ⋅ = + ⋅ + a b c a b a c ( ) ( ) ( )? Give an example to support your answer. No. + ⋅ ≠ + ⋅ + 2 (3 4) (2 3) (2 4). In Exercises 41–52, state the name of the property illustrated. 41. + = ⋅ + ⋅ x x 7( 2) 7 7 2 Distributive property 42. + = + 15 9 9 15 Commutative property of addition 43. ⋅ ⋅ = ⋅ ⋅ (78)9 7(89) Associative property of multiplication 44. + = + c d d c Commutative property of addition 45. + + = + + (24 7) 3 24 (7 3) Associative property of addition 46. ⋅ ⋅ = ⋅ ⋅ x x 4(11 ) (411) Associative property of multiplication 47. ⋅ = ⋅ 3 7 7 3 Commutative property of multiplication 48. + ⎛ + ⎝⎜ ⎞ ⎠⎟ = ⎛ + ⎝⎜ ⎞ ⎠⎟ + 3 8 1 8 3 2 3 8 1 8 3 2 Associative property of addition 49. − + = − ⋅ + − ⋅ x x 1( 4) ( 1) ( 1) 4 Distributive property 50. + ⋅ = ⋅ + ⋅ r s t r t s t ( ) ( ) ( ) Distributive property 51. + + = + + r s t t r s ( ) ( ) Commutative property of addition 52. ⋅ + = + ⋅ g h i h i g ( ) ( ) Commutative property of multiplication In Exercises 53– 64, use the distributive property to multiply. Then, if possible, simplify the resulting expression. 53. + b 2( 5) + b2 10 54. + a 8( 6) + a8 48 55. − c 3( 7) − c3 21 56. − d 6( 2) − d6 12 57. − − x (5 1) − + x5 1 58. − − x 9(6 2) − + x 54 18 59. ⎛ − ⎝⎜ ⎞ ⎠⎟ x 32 1 16 1 32 − x2 1 60. ⎛ − ⎝⎜ ⎞ ⎠⎟ x 15 2 3 4 5 − x 10 12 61. − 2( 8 2) 2 62. − − 3(2 3) 6 3 3 − + 63. + 5( 2 3) + 5 2 5 3 64. − 5( 15 20) − 5 3 10 Challenge Problem/Group Activity 65. Does ÷ = ÷ a a 0 0 (assume ≠ a 0)? Explain. No. ÷ = a 0 0 (when ≠ a 0), but ÷ a 0 is undefined. Recreational Mathematics 66. KenKen Refer to the Recreational Math box on page 260. Complete the following KenKen puzzle. * 67. a) Consider the three words man eating tiger. Does (man eating) tiger mean the same as man (eating tiger)? No b) Does (horse riding) monkey mean the same as horse (riding monkey)? No c) Can you find three other nonassociative word triples? Answers will vary. Research Activity 68. Complex Numbers A set of numbers that was not discussed in this chapter is the set of complex numbers. Write a report on complex numbers. Include their relationship to the real numbers. 2 24 22 32 12 243 51 71 *See Instructor Answer Appendix
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