89 R.8 Radical Expressions Rationalizing Denominators Condition 3 for a simplified radical requires that no denominator contain a radical. We achieve this by rationalizing the denominator—that is, multiplying by a form of 1. (c) A 22 + 3B A 28 - 5B = 22 A 28 B - 22 152 + 328 - 3152 FOIL method = 216 - 522 + 3 A222 B - 15 Multiply; 28 = 222 = 4 - 522 + 622 - 15 Simplify. = -11 + 22 Combine like terms. S Now Try Exercises 117 and 125. EXAMPLE 10 Rationalizing Denominators Rationalize each denominator. (a) 42 3 (b) B4 3 5 SOLUTION (a) 42 3 = 42 3 # 23 23 Multiply by 23 23 (which equals 1). = 423 3 In the denominator, 2a # 2a = a. (b) B4 3 5 = 24 3 24 5 Quotient rule The denominator will be a rational number if it equals 24 54. That is, four factors of 5 are needed under the radical. We multiply by 24 53 2 4 53 . 2 4 32 4 5 Because 24 5 has just one factor of 5, three additional factors are needed. = 24 3 # 24 53 24 5 # 24 53 Multiply by 24 53 2 4 53 . = 24 3 # 53 24 54 Product rule = 24 375 5 Simplify. S Now Try Exercises 127 and 131.
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