41 R.4 Integer and Rational Exponents Power Rules for Exponents For all integers m and n and all real numbers a and b, the following rules hold true. Rule Example Description Power Rule 1 1 am2n =amn 14523 = 45 # 45 # 45 = 45+5+5 = 45 # 3 = 415 To raise a power to a power, multiply the exponents. Power Rule 2 1 ab2m =ambm 17x23 = 17x217x217x2 = 17 # 7 # 721x # x # x2 = 73x3 To raise a product to a power, raise each factor to that power. Power Rule 3 aa bb m = am bm 1 b 302 a3 5b 4 = a 3 5ba 3 5ba 3 5ba 3 5b = 3 # 3 # 3 # 3 5 # 5 # 5 # 5 = 34 54 To raise a quotient to a power, raise the numerator and the denominator to that power. EXAMPLE 2 Using the Power Rules Simplify. Assume all variables represent nonzero real numbers. (a) 15322 (b) 134x223 (c) a 25 b4b 3 (d) a -2m6 t2z b 5 SOLUTION (a) 15322 = 53122 = 56 Power rule 1 (b) 134x223 = 134231x223 Power rule 2 = 34132x2132 Power rule 1 = 312x6 (c) a 25 b4b 3 = 12523 1b423 Power rule 3 = 215 b12 Power rule 1 (d) a -2m6 t2z b 5 = 1-2m625 1t2z25 Power rule 3 = 1-2251m625 1t225z5 Power rule 2 = -32m30 t10z5 Evaluate 1-225. Then use Power rule 1. = - 32m30 t10z5 -a b = - a b S Now Try Exercises 23, 25, 29, and 31.
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