SECTION A.10 nth Roots; Rational Exponents A89 3 Rationalize Denominators and Numerators When radicals occur in quotients, it is customary to rewrite the quotient so that the new denominator contains no radicals. This process is referred to as rationalizing the denominator . The idea is to multiply by an appropriate expression so that the new denominator contains no radicals. For example: If a Denominator Contains the Factor Multiply by To Obtain a Denominator Free of Radicals 3 3 3 3 2 ( ) = 3 1 + 3 1 − 3 1 3 1 2 2 2 ( ) − = − = 2 3 − 2 3 + 2 3 2 9 7 2 2 ( ) − = − = − 5 3 − 5 3 + 5 3 5 3 2 2 2 ( ) ( ) − = − = 4 3 2 3 4 2 8 2 3 3 3 ⋅ = = In rationalizing the denominator of a quotient, be sure to multiply both the numerator and the denominator by the expression. Now Work problem 59 In calculus, it is sometimes necessary to rationalize the numerator of a quotient. To do this, multiply by an appropriate expression so that the new numerator contains no radicals. Rationalizing Denominators Rationalize the denominator of each expression: (a) 1 3 (b) 5 4 2 3 (c) 2 3 3 2 − Solution EXAMPLE 5 (a) The denominator contains the factor 3, so we multiply the numerator and denominator by 3 to obtain 1 3 1 3 3 3 3 3 3 3 2 ( ) = ⋅ = = (b) The denominator contains the factor 2, 3 so we multiply the numerator and denominator by 4 3 to obtain ( ) = ⋅ = = ⋅ = 5 4 2 5 4 2 4 4 5 4 4 8 5 4 4 2 5 4 8 3 3 3 3 3 3 3 3 (c) The denominator contains the factor 3 3 2, − so we multiply the numerator and denominator by 3 3 2 + to obtain 2 3 3 2 2 3 3 2 3 3 2 3 3 2 2 3 3 2 3 3 2 2 3 3 2 3 18 6 6 15 6 6 15 2 2 2 ( ) ( ) ( ) ( ) − = − ⋅ + + = + − = + − = + − = − +
RkJQdWJsaXNoZXIy NjM5ODQ=