44 CHAPTER R Review of Basic Concepts We have shown several examples where the product and power rules were used to simplify expressions with positive integer exponents. These rules continue to apply in expressions involving negative exponents, as seen in the next example. CAUTION When applying the quotient rule, be sure to subtract the exponents in the correct order. Be especially careful to avoid sign errors when the exponent in the denominator is negative. EXAMPLE 5 Using the Quotient Rule Simplify each expression. Assume all variables represent nonzero real numbers. (a) 125 122 (b) a5 a-8 (c) 16m-9 12m11 (d) 25r7z5 10r9z SOLUTION (a) 125 122 = 125-2 = 123 (b) a5 a-8 = a5-1-82 = a13 Use parentheses to avoid errors. (c) 16m-9 12m11 = 16 12 # m-9-11 = 4 3 m-20 = 4 3 # 1 m20 = 4 3m20 (d) 25r7z5 10r9z = 25 10 # r7 r9 # z5 z1 = 5 2 r7-9z5-1 = 5 2 r-2z4 = 5z4 2r2 S Now Try Exercises 49, 55, 57, and 59. EXAMPLE 6 Using the Rules for Exponents Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. (a) 3x-214-1x-522 (b) 12p3q-1 8p-2q (c) 13x22-113x52-2 13-1x-222 SOLUTION (a) 3x-214-1x-522 = 3x-214-2x-102 Power rules = 3 # 4-2 # x-2+1-102 Rearrange factors; product rule = 3 # 4-2 # x-12 Simplify the exponent on x. = 3 16x12 Write with positive exponents, and multiply.
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