78 CHAPTER R Review of Basic Concepts (Method 1) = a a + 1 1a21a + 12 + 1 a 1a21a + 12 1 a 1a21a + 12 + 1 a + 1 1a21a + 12 Distributive property = a2 + 1a + 12 1a + 12 + a Multiply. = a2 + a + 1 2a + 1 Combine like terms. a a + 1 + 1 a 1 a + 1 a + 1 = a2 + 11a + 12 a1a + 12 11a + 12 + 11a2 a1a + 12 For Method 2, find the LCD, and add terms in the numerator and denominator of the complex fraction. (Method 2) (b) a a + 1 + 1 a 1 a + 1 a + 1 = a a a + 1 + 1 aba1a + 12 a1 a + 1 a + 1ba1a + 12 The result is the same as in Method 1. For Method 1, multiply both numerator and denominator by the LCD of all the fractions, a1a + 12. = a2 + a + 1 a1a + 12 2a + 1 a1a + 12 Combine terms in the numerator and denominator. = a2 + a + 1 a1a + 12 # a1a + 12 2a + 1 Multiply by the reciprocal of the divisor. = a2 + a + 1 2a + 1 Multiply fractions, and write in lowest terms. S Now Try Exercises 71 and 83. Negative exponents are sometimes used to write complex fractions. Recall that for any nonzero real number a and integer n, a−n = 1 an . Definition of negative exponent EXAMPLE 6 Simplifying a Rational Expression with Negative Exponents Simplify 1x + y2-1 x-1 + y-1 . Write the result with only positive exponents. SOLUTION 1x + y2-1 x-1 + y-1 = 1 x + y 1 x + 1 y Definition of negative exponent
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