8.2.1 Radical Expressions and Rational Exponents Key Objectives • Rewrite radical expressions by using rational exponents. • Simplify and evaluate radical expressions and expressions containing rational exponents. Key Terms • In the radical x n , which represents the n th root of x , n is the index . A radical that shows no index is a square root (where the index equals 2). • A rational exponent is an exponent that can be expressed as m / n such that if m and n are integers and n ≠ 0, then b b b ( ) m n n m n m = = . Example 1 Finding Real Roots Recall that squaring and taking the square root are inverse operations. Similarly, there are roots that correspond to larger powers. Generally, a is the n th root of b if a n = b . Therefore, for b n where n is a positive integer: • If n is odd, then there is one real n th root of b .
• If n is even and b > 0, then there are two real n th roots of b . • If n is even and b < 0, then there are no real n th roots of b . If b = 0, then the one real n th root of b is 0.
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