Algebra 2: Unit #6 Radical Functions (6.4 NOTES)
6.4 Rational Exponents
Name: ___________________________
Block: ___________
Big Idea: You can write a radical expression in an equivalent from using a rational (fraction) exponent.
𝑚
𝑎 𝑛 = _____________ = ______________
To simplify/evaluate:
𝑛
Method #1: evaluate √𝑎 , then raise to the power of m
Denominator tells me _________________________
Numerator tells me ___________________________
2
3
√𝑥 = _____
4
√𝑥 = _____
√𝑥 = ____
Method #2: evaluate 𝑎𝑚 , then take the nth root of that value
5
√𝑥 = ____
Part I: Use the definition of rational exponents to rewrite the radical expression using rational exponents.
a)  10 
6
3
4
b) √2
 2
4
c)
5
 5
5
d)
4
e)
3
2
Use the definition of rational exponents to rewrite the expression using radical notation. Then, simplify if possible.
4
3
a) 273
3
b) 42
2
c) 1002
3
3
d) (−8)
e)
x5
Part II: Simplify without a calculator. (All answers should be rational numbers!) Think about how converting
between rational and radical form can help! **Remember the negative exponent property!!
3
2
3
a) (−32)5
c) (16)−2
b) (250)3
−3
3
d) 10000 4
e)
4 −2
( )
9
Part III: Multiplying and Dividing “Un-Like” Radicals – Use rational exponents to find a common root
(common denominator) then simplify (properties of exponents)! Give your answer in rational AND radical notation.
6
√𝑥
3
a) √𝑥 ∙ √𝑥
2
b) 3
c)
√𝑥
5
1
d) (2𝑥 3 ) (∙ −5𝑥 4 )
e)
−100 √𝑥 3
20√𝑥
10𝑥 2 √𝑚
4
2𝑥 3 √𝑚
1
1
f) (16𝑥 2 𝑦 3 )4 (8𝑥 3 𝑦 9 )3
Part III: Properties of Rational Exponents: the same rules apply when simplifying rational expressions!
Simplify the following expressions using the properties of exponents. Give your answer in simplified radical
AND rational exponent form if possible. Rationalize all denominators.
3
a)
5
4
(𝑛4 )2

1
4
c) (−8𝑥√𝑥𝑦)
3
d) (
4
3 √𝑥
2
)
(16𝑏12 )
3
1 3
(−12𝑝2 )−2
2
3
4
(9𝑥 √𝑦)
3
2
𝑎8
(3𝑥 )(2𝑥 )
3
 

b) 16 y 5 


2 √𝑥
1
2
(36𝑎2 )2
3
2
3
√−27𝑥
6
√𝑥 5
2
2
[(−27𝑥 2 𝑦 9 )3 ][(8𝑥 6 )]3
1
10
3
(√(25𝑥)4 ) (𝑥 2 )
2