45 R.4 Integer and Rational Exponents (b) 12p3q-1 8p-2q = 12 8 # p3 p-2 # q-1 q1 Write as separate factors. = 3 2 # p3-1-22q-1-1 Write 12 8 in lowest terms; quotient rule = 3 2 p5q-2 Simplify the exponents. = 3p5 2q2 Write with positive exponents, and multiply. (c) 13x22-113x52-2 13-1x-222 = 3-1x-23-2x-10 3-2x-4 Power rules = 3-1+1-22x-2+1-102 3-2x-4 Product rule = 3-3x-12 3-2x-4 Simplify the exponents. = 3-3-1-22x-12-1-42 Quotient rule = 3-1x-8 Simplify the exponents. = 1 3x8 Write with positive exponents, and multiply. S Now Try Exercises 63, 69, and 71. Be careful with signs. CAUTION Notice the use of power rule 2, 1ab2m = ambm, in Example 6(c). Remember to apply the exponent to the numerical coefficient 3. 13x22-1 = 3-11x22-1 = 3-1x-2 Rational Exponents The definition of an can be extended to rational values of n by defining a1/n to be the nth root of a. By power rule 1 (extended to a rational exponent), we have the following. 1a1/n2n = a11/n2n = a1 = a This suggests that a1/n is a number whose nth power is a. The Expression a1/n a1/n, n Even If n is an even positive integer, and if a 70, then a1/n is the positive real number whose nth power is a. That is, 1a1/n2n = a. (In this case, a1/n is the principal nth root of a.) a1/n, n Odd If n is an odd positive integer, and a is any nonzero real number, then a1/n is the positive or negative real number whose nth power is a. That is, 1a1/n2n = a. For all positive integers n, 01/n = 0.
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