By Constance Reid

This energetic advisor through a admired historian specializes in the function of Euclid's *Elements* in mathematical advancements of the final 2,000 years. No mathematical heritage past straightforward algebra and aircraft geometry is important to understand the clear and straightforward factors, that are augmented by means of greater than eighty drawings. 1963 variation.

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4a, or a # 4 47. Let p represent Panya’s speed and w the wind speed; p - 2w 55. Yes 63. Let x represent the unknown number; 7x = 1596 69. 334w = 85 79. 2 The Commutative, Associative, and Distributive Laws 15 Collaborative Corner Teamwork Focus: Group problem solving; working collaboratively Time: 15 minutes Group Size: 2 Are two (or more) heads better than one? Try this activity and decide whether group work can help us solve problems. ACTIVITY 1. The left-hand column below contains the names of 12 colleges.

For any numbers a, b, and c, a # 1b # c2 = 1a # b2 # c. ) EXAMPLE 2 Use an associative law to write an expression equivalent to each of the following: (a) y + 1z + 32; (b) 18x2y. SOLUTION a) y + 1z + 32 is equivalent to 1y + z2 + 3 by the associative law of addition. b) 18x2y is equivalent to 81xy2 by the associative law of multiplication. Try Exercise 33. When only addition or only multiplication is involved, parentheses do not change the result. For that reason, we sometimes omit them altogether.

7 5 a a 4 3 63. + a a 65. 8 3 + 10 15 66. 5 7 + 8 12 67. 9 2 7 7 68. 2 12 5 5 69. 13 4 18 9 70. 13 8 15 45 71. 2 20 30 3 72. 5 5 7 21 73. 7 3 , 6 5 74. 7 3 , 5 4 75. 8 4 , 9 15 76. 1 1 , 8 4 3 7 78. 10 , 10 9 79. 7 7 , 13 13 5 17 , 8 6 81. 82. 3 8 1 5 77. 12 , 80. 83. 9 1 2 84. 2 7 5 3 3 7 ! Aha Fraction Notation 91. In the table below, the top number can be factored in such a way that the sum of the factors is the bottom number. For example, in the first column, 56 is factored as 7 # 8, since 7 + 8 = 15, the bottom number.