85 R.8 Radical Expressions The following rules for working with radicals are simply the power rules for exponents written in radical notation. NOTE When working with variable radicands, we will usually assume that all variables in radicands represent only nonnegative real numbers. Rules for Radicals Suppose that a and b represent real numbers, and m and n represent positive integers for which the indicated roots are real numbers. Rule Description Product rule !n a # !n b =!n ab The product of two roots is the root of the product. Quotient rule Ån a b =!n a!n b 1 b 302 The root of a quotient is the quotient of the roots. Power rule !m !n a = "mn a The index of the root of a root is the product of their indexes. EXAMPLE 5 Using the Rules for Radicals Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. (a) 26 # 254 (b) 23 m # 23 m2 (c) 27 # 211xy (d) B7 64 (e) B4 a b4 (f) 27 1 3 2 (g) 24 13 (h) 25 # 23 2 SOLUTION (d) B7 64 = 27 264 = 27 8 Quotient rule (e) B4 a b4 = 24 a 24 b4 = 24 a b (f) 27 1 3 2 = 221 2 Power rule (h) 25 # 23 2 This product cannot be found using the product rule for radicals because the indexes are different. S Now Try Exercises 47, 51, 55, 59, and 69. (a) 26 # 254 = 26 # 54 Product rule = 2324 Multiply. = 18 (b) 23 m # 23 m2 = 23 m3 = m (c) 27 # 211xy = 277xy (g) 24 13 = 23 4# 2 = 28 3
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