1. 7-4 Rational Exponents
Hubarth
Algebra II

2. Ex. 1 Simplify Expressions With Rational Exponents
Simplify each expression.
a b ∙ c ∙
3 125
5 ∙ 5
3 10 ∙ 3 100 5 25
3 1000 5 10

3. Rational Exponent
If the nth root of a is a real number and m is an integer , then
𝑎 1 𝑛 = 𝑛 𝑎 and 𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 =( 𝑛 𝑎 ) 𝑚
Ex. 2 Converting to and From Radical Form
a. Write the exponential expressions 𝑥 and 𝑦 −2.5 in radical form
5 𝑥 3
𝑦 −2.5 = 𝑦 −5 2
( 5 𝑥 ) 3
1 𝑦 5 = 1 ( 𝑦 ) 5
b. Write the radical expressions 𝑎 3 and ( 5 𝑏 ) 2
𝑎 3
( 5 𝑏 ) 2
𝑎 3 2
𝑏 2 5

4. Summary Properties of Rational Exponents
Let m and n represent rational numbers. Assume that no denominator equals 0.
Property Example
𝑎 𝑚 ∙ 𝑎 𝑛 = 𝑎 𝑚+𝑛 ∙ = = 8 1 =8
( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚𝑛 ( ) 4 = ∙4 = 5 2 =25
(𝑎𝑏 ) 𝑚 = 𝑎 𝑚 ∙ 𝑏 𝑚 (4∙5 ) = ∙ =2∙
𝑎 −𝑚 = 1 𝑎 𝑚 − 1 2 = = 1 3
𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛 𝜋 𝜋 = 𝜋 3 2 − 1 2 = 𝜋 1 =𝜋
( 𝑎 𝑏 ) 𝑚 = 𝑎 𝑚 𝑏 𝑚 ( 5 27 ) = =

5. Ex. 3 Simplifying Numbers with Rational Exponents
Simplify each number.
a. (−32 ) b. 4 −3.5
( 5 −32 ) 3
4 − 7 2
( 5 (− 2) 5 ) 3
(−2 ) 3
1 ( 4 ) 7 −8
1 2 7
1 128

6. Ex. 4 Writing Equation in Simplest Form
Write (16 𝑦 −8 ) − in simplest form.
(16 𝑦 −8 ) − 3 4 =( 16 𝑦 8 ) − 3 4
( 𝑦 8 ) −3
( 2 𝑦 2 ) −3
( 𝑦 ) 3
𝑦 6 8

7. Practice
Simplify each expression.
a b ∙
2 ∙ 8
4 16 2
16 =4
2. Write the expressions 𝑦 − and 𝑧 0.4 in radical form.
1 8 𝑦 3
𝑧 0.4 = 𝑧 2 5
5 𝑧 2
3. Write the expressions 3 𝑥 2 and ( 𝑦 ) 3 in exponential form.
𝑥 2 3
𝑦 3 2
4. Simplify each number.
a. 25 − b. (−32 ) 4 5
( 25 ) −3
( 5 −32 ) 4
= 1 125
(−2 ) 4 =16
5. Write (8 𝑥 15 ) − in simplest form
( 3 8 𝑥 15 ) −1
(2 𝑥 5 ) −1 = 1 2 𝑥 5