1. Radical Functions and Rational Exponents
Chapter 6
Radical Functions and Rational Exponents

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

3. 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

4. Example 1 Simplifying Expressions with Rational Exponents
1251/3 5½
2½ ½
2½ ½
P/R = power/root r
√x p
( √x )p r * *

5. 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

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

7. Properties of Rational Exponents
Let m and n represent rational numbers. Assume that no denominator = 0.
Property Example
am * an = a m+n ⅓ * 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

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

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

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