33 R.3 Real Number Operations and Properties Figure 18 helps to explain the distributive property. The area of the entire region shown can be found in two ways, as follows. 415 + 32 = 4182 = 32 or 4152 + 4132 = 20 + 12 = 32 The result is the same. This means that 415 + 32 = 4152 + 4132. EXAMPLE 7 Simplifying Expressions Use the commutative and associative properties to simplify each expression. (a) 6 + 19 + x2 (b) 5 8 116y2 (c) -10pa 6 5b SOLUTION (a) 6 + 19 + x2 = 16 + 92 + x Associative property = 15 + x Add. (b) 5 8 116y2 = a 5 8 # 16by Associative property = 10y Multiply. (c) -10pa 6 5b = 6 5 1-10p2 Commutative property = c 6 5 1-102d p Associative property = -12p Multiply. S Now Try Exercises 127 and 129. 5 3 4 Geometric Model of the Distributive Property Figure 18 EXAMPLE 8 Using the Distributive Property Rewrite each expression using the distributive property and simplify, if possible. (a) 31x + y2 (b) -1m- 4n2 (c) 7p + 21 (d) 1 3 a 4 5 m3 2 n - 27b SOLUTION (a) 31x + y2 = 3x + 3y Distributive property (b) -1m- 4n2 = -11m- 4n2 = -11m2 + 1-121-4n2 = -m+ 4n Be careful with the negative signs. (c) 7p + 21 = 7p + 7 # 3 = 7 # p + 7 # 3 = 71p + 32 Distributive property in reverse (d) 1 3 a 4 5 m3 2 n - 27b = 1 3 a 4 5 mb + 1 3 a- 3 2 nb + 1 3 1-272 = 4 15 m1 2 n - 9 S Now Try Exercises 131 and 133.
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