SECTION A.10 nth Roots; Rational Exponents A91 DEFINITION a n1 If a is a real number and n 2 ≥ is an integer, then a a n n 1 = (3) provided that a n exists. DEFINITION am n If a is a real number and m and n are integers containing no common factors, with n 2, ≥ then a a a m n n m n m ( ) = = (4) provided that a n exists. 5 Simplify Expressions with Rational Exponents Radicals are used to define rational exponents. Note that if n is even and a 0 < , then a n and a n 1 do not exist. We have two comments about equation (4): • The exponent m n must be in lowest terms, and n 2 ≥ must be positive. • In simplifying the expression a , m n either a n m or a n m ( ) may be used, the choice depending on which is easier to simplify. Generally, taking the root first, as in a , n m ( ) is easier. Now Work problem 81 It can be shown that the Laws of Exponents hold for rational exponents. The next example illustrates using the Laws of Exponents to simplify. Simplifying Expressions With Rational Exponents Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. (a) x y x y 2 3 2 1 2 ( )( ) − (b) x y 2 1 3 2 3 3 − (c) x y x y 9 2 1 3 1 3 1 2 EXAMPLE 10 Simplifying Expressions With Rational Exponents (a) 4 4 2 8 3 2 3 3 ( ) = = = (b) 8 8 2 16 4 3 3 4 4 ( ) ( ) ( ) − = − = − = (c) 32 32 2 1 4 2 5 5 2 2 ( ) ( ) = = = − − − (d) 25 25 25 5 125 6 4 3 2 3 3 ( ) = = = = EXAMPLE 9 Writing Expressions Containing Fractional Exponents as Radicals (a) 4 4 2 1 2 = = (b) 8 8 2 2 1 2 = = (c) 27 27 3 1 3 3 ( ) − = − = − (d) 16 16 2 2 1 3 3 3 = = EXAMPLE 8 (continued)
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