SECTION A.10 nth Roots; Rational Exponents A87 A. 10 n th Roots; Rational Exponents PREPARING FOR THIS SECTION Before getting started, review the following: • Exponents, Square Roots (Section A.1, pp. A7–A10) Now Work the ‘Are You Prepared?’ problems on page A93. OBJECTIVES 1 Work with n th Roots (p. A87) 2 Simplify Radicals (p. A88) 3 Rationalize Denominators and Numerators (p. A89) 4 Solve Radical Equations (p. A90) 5 Simplify Expressions with Rational Exponents (p. A91) 1 Work with n th Roots DEFINITION Principal n th Root The principal n th root of a real number a , n 2 ≥ an integer, symbolized by a, n is defined as follows: a b a b means n n = = where a 0 ≥ and b 0 ≥ if n is even and a , b are any real numbers if n is odd. Notice that if a is negative and n is even, then a n is not defined. When it is defined, the principal n th root of a number is unique. The symbol a n for the principal n th root of a is called a radical ; the integer n is called the index , and a is called the radicand . If the index of a radical is 2, we call a 2 the square root of a and omit the index 2 by simply writing a. If the index is 3, we call a 3 the cube root of a . These are examples of perfect roots , since each simplifies to a rational number. Notice the absolute value in Example 1(d). If n is even, then the principal n th root must be nonnegative. In general, if n 2 ≥ is an integer and a is a real number, we have a a n if 3 is odd n n = ≥ (1a) a a n if 2 iseven n n = ≥ (1b) Now Work problem 11 Radicals provide a way of representing many irrational real numbers. For example, it can be shown that there is no rational number whose square is 2. Using radicals, we can say that 2 is the positive number whose square is 2. In Words The symbol a n means “what is the number that, when raised to the power n , equals a .” Simplifying Principal n th Roots (a) 8 2 2 3 3 3 = = (b) 64 4 4 3 3 3 ( ) − = − = − (c) 1 16 1 2 1 2 4 4 4 ( ) = = (d) 2 2 2 6 6 ( ) − = − = EXAMPLE 1
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