1. Properties of Rational Exponents and Radicals
Section 5.2 beginning on page 243

2. Properties of Rational Exponents
Note: You have already learned and used all of these properties.
Product of Powers: 𝑎 𝑚 ∙ 𝑎 𝑛 = 𝑎 𝑚+𝑛
Power of a Power: ( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚∙𝑛
Power of a Product: (𝑎𝑏) 𝑚 = 𝑎 𝑚 𝑏 𝑚
Negative Exponent: 𝑎 −𝑚 = 1 𝑎 𝑚 , 𝑎≠0
Zero Exponent: 𝑎 0 =1, 𝑎≠0
Quotient of Powers: 𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛 , 𝑎≠0
Power of a Quotient: 𝑎 𝑏 𝑚 = 𝑎 𝑚 𝑏 𝑚 , 𝑏≠0

3. Using These Properties
Example 1: Simplify each expression.
a) ∙
b) ∙
c) ∙ −1 5 d) e)
= ∙ =
= ∙
=6∙
= 1 4∙3
= 1 12
= 4 −5 5 ∙ 3 −5 5
= 4 −1 ∙ 3 −1
= 5 1 ∙ 5 −1 3
= ∙ 5 −1 3 = = =
=7 2 3

4. Simplifying Radical Expressions
Example 2:
3 12 ∙ 3 18
= 3 12∙18
= 3 216 =6 =
= 4 16 =2

5. Simplest Radical Form An expression is in simplest form when:
No radicands have a perfect nth power as a factor
No radicands contain fractions
No radicals appear in the denominator of a fraction
Example 3: Write each expression in simplest radical form.
a) b)
Find a perfect cube factor of 135… ∙ = =
= 3 27 ∙ 3 5
Find the number you can multiply 8 by and get a perfect 5th power...
=3 3 5

6. Simplest Radical Form 1 5+ 3 ∙ (5− 3 ) (5− 3 ) = 5− 3 25−3 = 5− 3 22
If a denominator has a sum or difference involving square roots, multiply the numerator and the denominator by the conjugate of the denominator.
Example 4: Write in simplest form.
The conjugate of is 5− 3
∙ (5− 3 ) (5− 3 )
= 5− −3
= 5−
(5+ 3 )(5− 3 )
=25− −3
=25−3

7. Simplest Radical Form
Radical expressions with the same index and radicand are like radicals. To add or subtract like radicals, we use the distributive property. (It is like combining like terms)
Example 5: Simplify each expression.
a) b) 2 ( )+10 ( ) c) − 3 2
=(1+7) 4 10
=(2+10) ( )
= 3 27 ∙ 3 2 − 3 2
=12 ( )
=9 3 2 − 3 2
=8 4 10
=(9−1) 3 2
=8 3 2

8. Simplifying Variable Expressions
The properties of rational exponents and radicals can also be applied to expressions involving variables. Because a variable can be positive, negative, or zero, sometimes absolute value is needed when simplifying a variable expression.
Example 6: Simplify each expression
a) 𝑦 b) 4 𝑥 4 𝑦 8
= 4 𝑥 𝑦 8
= 4 𝑥 ( 𝑦 2 ) 4
= 𝑥 𝑦 2
= ( 𝑦 2 ) 3
=4 𝑦 2

9. Writing Variable Expressions in Simplest form
Example 7: Write each expression in simplest form. Assume all variables are positive.
a) 5 4 𝑎 8 𝑏 14 𝑐 5 b) 𝑥 3 𝑦 c) 14𝑥 𝑦 𝑥 𝑧 −6
∙ 3 𝑦 3 𝑦
= 5 𝑎 5 𝑏 10 𝑐 5 ∙ 5 4 𝑎 3 𝑏 4
= 𝑥 3 𝑦 3 𝑦 9
=7 𝑥 1− 𝑦 𝑧 6
=𝑎 𝑏 2 𝑐 5 4 𝑎 3 𝑏 4
=7 𝑥 𝑦 𝑧 6
= 𝑥 3 𝑦 𝑦 3

10. Adding and Subtracting Variable Expressions
Example 8: Perform each indicated operation. Assume all variables are positive.
a) 5 𝑦 +6 𝑦 b) 𝑧 5 −𝑧 3 54 𝑧 2
= 𝑧 3 𝑧 2 −𝑧 3 27∙2 𝑧 2
=(5+6) 𝑦
=11 𝑦
=12𝑧 3 2 𝑧 2 −3𝑧 3 2 𝑧 2
=(12𝑧−3𝑧) 3 2 𝑧 2
=9𝑧 3 2 𝑧 2

11. Monitoring Progress Simplify the expression.
∙ ) ) ) ∙
5) ∙ ) ) ) 𝑥 10 𝑥 5
9) 3 6− ) − 5 12
11) ( ) 12) = = =8
=5∙ =
=2 3 13 =3 =5 =
=6 5 12 =
=3 3 5

12. Monitoring Progress
Simplify the expression. Assume all variables are positive.
13) 𝑞 ) 5 𝑥 10 𝑦 ) 6𝑥 𝑦 𝑥 𝑦 ) 9 𝑤 5 −𝑤 𝑤 3
= 𝑥 2 𝑦
=3 𝑞 3
=2 𝑥 𝑦 1 4
=3 𝑤 2 𝑤 − 𝑤 2 𝑤
=2 𝑤 2 𝑤