43 R.4 Integer and Rational Exponents Example 4(c) showed the following. a2 5b -3 = 125 8 = a 5 2b 3 We can generalize this result. If a≠0 and b≠0, then for any integer n, the following holds true. aa bb −n = a b ab n The following example suggests the quotient rule for exponents. 56 52 = 5 # 5 # 5 # 5 # 5 # 5 5 # 5 = 54 This exponent is the result of dividing out common factors, or, essentially, subtracting the original exponents. EXAMPLE 4 Using the Definition of a Negative Exponent Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. (a) 4-2 (b) -4-2 (c) a 2 5b -3 (d) 1xy2-3 (e) xy-3 SOLUTION (a) 4-2 = 1 42 = 1 16 (b) -4-2 = - 1 42 = - 1 16 (c) a 2 5b -3 = 1A 2 5B 3 = 1 8 125 = 1 , 8 125 = 1 # 125 8 = 125 8 Multiply by the reciprocal of the divisor. (d) 1xy2-3 = 1 1xy23 = 1 x3y3 (e) xy-3 = x # 1 y3 = x y3 Base is xy. Base is y. S Now Try Exercises 37, 39, 41, and 43. CAUTION A negative exponent indicates a reciprocal, not a sign change of the expression. Quotient Rule Let m and n be integers and a be a nonzero real number. am an =am−n That is, when dividing powers of like bases, keep the same base and subtract the exponent of the denominator from the exponent of the numerator.
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