Radical Functions and Rational Exponents Chapter 6 Radical Functions and Rational Exponents

To simplify expressions with rational exponents What you’ll learn … To simplify expressions with rational exponents

Another way to write a radical expression is to use a rational exponent. Like the radical form, the exponent form always indicates the principal root. √25 = 25½ 3 √27 = 27⅓ 4 √16 = 161/4

Example 1 Simplifying Expressions with Rational Exponents 1251/3 5½ 2½ 2½ 2½ 8½ P/R = power/root r √x p ( √x )p r * *

A rational exponent may have a numerator other than 1 A rational exponent may have a numerator other than 1. The property (am)n = amn shows how to rewrite an expression with an exponent that is an improper fraction. Example 253/2 = 25(3*1/2) = (253)½ = √253

Example 2 Converting to and from Radical Form y -2.5 y -3/8 √a3 ( √b )2 √x2 5 3

Properties of Rational Exponents Let m and n represent rational numbers. Assume that no denominator = 0. Property Example am * an = a m+n 8⅓ * 8⅔ = 8 ⅓+⅔ = 81 =8 (am)n = amn (5½)4 = 5½*4 = 52 = 25 (ab)m = ambm (4 *5)½ = 4½ * 5½ =2 * 5½

Properties of Rational Exponents Let m and n represent rational numbers. Assume that no denominator = 0. Property Example a-m 1 9 -½ 1 1 am 9 ½ 3 am a m-n π3/2 π 3/2-1/2 = π1 = π an π ½ a m am 5 5⅓ 5⅓ b bm 27 27 ⅓ 3 = = = = = ⅓ = =

Example 4 Simplifying Numbers with Rational Exponents (-32)3/5 4 -3.5

Example 5 Writing Expressions in Simplest Form (16y-8) -3/4 (8x15)-1/3