A92 APPENDIX Review Solution ab a b n n n ( ) = a a m n mn ( ) = a a a m n m n ⋅ = + a a 1 n n = − (a) x y x y x y x y x y x y x x y y x y y x 2 3 2 1 2 2 3 2 1 2 1 2 2 3 1 1 2 2 3 1 1 2 1 3 3 2 3 2 1 3 ( )( ) ( ) = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ = − − − − − (b) x y y x y x y x y x 2 2 2 2 8 1 3 2 3 3 2 3 1 3 3 2 3 3 1 3 3 2 3 1 3 3 2 ( ) ( ) ( ) ( ) = = = = − (c) x y x y x y x y x y x y 9 9 9 9 3 2 1 3 1 3 1 2 2 1 3 1 1 3 1 2 5 3 2 3 1 2 1 2 5 3 1 2 2 3 1/2 5 6 1 3 ( ) ( ) = = = = ( ) ( ) − − Now Work problem 101 The next two examples illustrate some algebra that you will need to know for certain calculus problems. Writing an Expression as a Single Quotient Write the following expression as a single quotient in which only positive exponents appear. x x x x 1 1 2 1 2 2 1 2 2 1 2 ( ) ( ) + + ⋅ + ⋅ − Solution EXAMPLE 11 x x x x x x x x x x x x x x x x 1 1 2 1 2 1 1 1 1 1 1 1 2 1 1 2 1 2 2 1 2 2 1 2 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + ⋅ + ⋅ = + + + = + + + + = + + + = + + − Now Work problem 107 Factoring an Expression Containing Rational Exponents Factor and simplify: x x x 4 3 2 1 2 1 3 4 3 ( ) + + Solution EXAMPLE 12 Begin by writing x2 4 3 as a fraction with 3 as the denominator. x x x x x x x x x x x x x x 4 3 2 1 2 4 2 1 3 6 3 4 2 1 6 3 2 2 2 1 3 3 2 7 2 3 1 3 4 3 1 3 4 3 1 3 4 3 1 3 1 3 ( ) ( ) ( ) ( ) [ ] ( ) + + = + + = + + = + + = + ↑ Add the two fractions. ↑ 2 and x1 3 are common factors. ↑ Simplify. Now Work problem 119
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