Property Product Rule Let a be any real number, and let m and n be positive integers. Then, am an amn To multiply expressions with the same base, keep the base the same and add the exponents. Find each product. a) 22 24 b) x9 x6 c) 5c3 7c9 d) (k)8 (k) (k)11 e) 43 52 Solution a) 22 24 224 26 64 Since the bases are the same, add the exponents. b) x9 x6 x96 x15 c) 5c3 7c9 (5 7)(c3 c9) Use the associative and commutative properties. 35c12 d) (k)8 (k) (k)11 (k)8111 (k)20 Product rule e) Can the product rule be applied to 43 52? No! The bases are not the same, so we cannot add the exponents. To evaluate 43 52, we evaluate 43 64 and 52 25, then multiply: 43 52 64 25 1600 If you do not see an exponent on the base, it is assumed that the exponent is 1. Look closely at part d). YOU TRY 2 Find each product. a) 3 32 b) y10 y4 c) 6m5 9m11 d) h4 h6 h4 e) 3372 3 Use the Power Rule (am)n amn What does (22)3 mean? We can rewrite (22)3 fi rst as 22 22 22. 22 22 22 2222 Use the product rule for exponents. 26 Add the exponents. 64 Simplify. Notice that (22)3 2222, or 22 3. This leads us to the basic power rule for exponents: When you raise a power to another power, keep the base and multiply the exponents. EXAMPLE 2 In-Class Example 2 Find each product. a) 52 5 b) y4 y9 c) 4x5 (10x8) d) d d7 d4 Answer: a) 125 b) y13 c) 40x13 d) d12 Property Basic Power Rule Let a be any real number, and let m and n be positive integers. Then, (am)n amn To raise a power to another power, keep the base the same and multiply the exponents. www.mhhe.com/messersmith SECTION 10.1 The Product Rule and Power Rules 771
messersmith_power_prealgebra_1e_ch4_7_10
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