EXAMPLE 5 In-Class Example 5 Simplify using the power rule for quotients. a) a4 5 b 3 b) ax 6 b 2 c) au v b 7 Answer: a) 64 125 b) x2 36 c) u7 v7 5 Use the Power Rule aa b b n an bn, where b 0 Another power rule allows us to simplify an expression like a 2 x b 4 . We can rewrite and simplify a 2 x b 4 as 2 x 2 x 2 x 2 x 2 2 2 2 x x x x 24 x4 16 x4 . To raise a quotient to a power, raise both the numerator and denominator to that power. Property Power Rule for a Quotient Let a and b be real numbers, and let n be a positive integer. Then, aa b b n an bn , where b 0 To raise a quotient to a power, raise both the numerator and denominator to that power. Simplify using the power rule for quotients. a) a3 8 b 2 b) a 5 x b 3 c) a t u b 9 Solution a) a3 8 b 2 32 82 9 64 b) a 5 x b 3 53 x3 125 x3 c) a t u b 9 t9 u9 YOU TRY 5 Simplify using the power rule for quotients. a) a 5 12 b 2 b) a2 d b 5 c) au v b 6 Let’s summarize the rules of exponents we have learned in this section: Summary The Product and Power Rules of Exponents In the rules below, a and b are any real numbers, and m and n are positive integers. Rule Example Product rule am an amn p4 p11 p411 p15 Basic power rule (am)n amn (c8)3 c83 c24 Power rule for a product (ab)n anbn (3z)4 34 z4 81z4 Power rule for a quotient aa b b n an w , (bn b 0) a2 b 4 w4 24 w4 16 www.mhhe.com/messersmith SECTION 10.1 The Product Rule and Power Rules 773
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