47 R.4 Integer and Rational Exponents The earlier results for integer exponents also apply to rational exponents. Definitions and Rules for Exponents Let r and s be rational numbers. For all positive numbers a and b, the following hold true. Product rule ar # as =ar+s Power rules 1ar2s =ars Quotient rule ar as =ar−s 1ab2r =arbr Negative exponent a−r = 1 ar a a bb r = ar br (b) 815/4 # 4-3/2 = 1811/425141/22-3 = 35 # 2-3 = 35 23 = 243 8 EXAMPLE 9 Using the Rules for Exponents Simplify each expression. Assume all variables represent positive real numbers. (a) 271/3 # 275/3 273 (b) 815/4 # 4-3/2 (c) 6y2/3 # 2y1/2 (d) a 3m5/6 y3/4 b 2a8y3 m6 b 2/3 SOLUTION (a) 271/3 # 275/3 273 = 271/3+5/3 273 Product rule = 272 273 Simplify. = 272-3 Quotient rule = 27-1 Simplify the exponent. = 1 27 Negative exponent (c) 6y2/3 # 2y1/2 = 6 # 2y2/3+1/2 Product rule = 12y7/6 Multiply. Simplify the exponent: 2 3 + 1 2 = 4 6 + 3 6 = 7 6 NOTE For all real numbers a, integers m, and positive integers n for which a1/n is a real number, am/n can be interpreted as follows. am/n = 1a1/n2m or am/n = 1am21/n So am/n can be evaluated either as 1a1/n2m or as 1am21/n. 274/3 = 1271/324 = 34 = 81 or 274/3 = 127421/3 = 531,4411/3 = 81 The result is the same.
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