Rational Exponents and Radical Expressions
Properties of Rational Exponents
Property Name
Definition
Product of Powers
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
Power of a Power
(𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
Power of a Product
(𝑎𝑏)𝑚 = 𝑎𝑚 ∙ 𝑏 𝑚
Negative Exponent
𝑎−𝑚 =
Example
5
1⁄
2
∙5
(3
5⁄ 2
2)
1⁄
2
(16 ∙ 9)
1
,𝑎 ≠ 0
𝑎𝑚
Zero Exponent
𝑎0 = 1, 𝑎 ≠ 0
Quotient of Powers
𝑎𝑚
= 𝑎𝑚−𝑛 , 𝑎 ≠ 0
𝑎𝑛
Power of a Quotient
𝑎
𝑎𝑚
( )𝑚 = 𝑚 , 𝑏 ≠ 0
𝑏
𝑏
3⁄
2
= 5(
36−
1⁄
2
= 52 = 25
5⁄ ∙2)
2
= 35 = 243
1⁄
2
1⁄
2
= 3(
= 16
1⁄ +3⁄ )
2
2
=
∙9
1
1
36 ⁄2
= 4 ∙ 3 = 12
=
1
6
2130 = 1
4
5⁄
2
1
4 ⁄2
= 4(
5⁄ −1⁄ )
2
2
= 42 = 16
1
27 1⁄
27 ⁄3 3
3
( ) = 1 =
64
64 ⁄3 4
Properties of Radicals
a and b are real numbers, n is an integer greater than 1
𝑛
Product Property
𝑛
𝑛
√𝑎 ∙ 𝑏 = √𝑎 ∙ √𝑏
Quotient Property
𝑛
𝑎 √𝑎
√ = 𝑛 ,𝑏 ≠ 0
𝑏 √𝑏
𝑛
A radical with index n is in simplest form when these three conditions are met.



No radicands have perfect nth powers as factors other than 1
No radicands contain fractions
No radicals appear in the denominator of a fraction
Because a variable can be positive, negative, or zero, sometimes absolute value is needed when simplifying a variable
expression.
Rule
When n is odd
When n is even
𝑛
√𝑥 𝑛 = 𝑥
𝑛
√𝑥 𝑛 = |𝑥|
Example
7
√57 = 5 and 7√(−5)7 = −5
4
√34 = 3 and 4√(−3)4 = 3
Transformation
Horizontal Translation
𝒇(𝒙) notation
Example
3
2 units right
3
3 units left
3
7 units up
3
1 unit down
3
In the y-axis
𝑔(𝑥) = √𝑥 − 2, ℎ(𝑥) = √𝑥 − 2
𝑓(𝑥 − ℎ)
Graph shifts left or right
𝑔(𝑥) = √𝑥 + 3, ℎ(𝑥) = √𝑥 + 3
Vertical Translation
𝑔(𝑥) = √𝑥 + 7, ℎ(𝑥) = √𝑥 + 7
𝑓(𝑥) + 𝑘
Graph shifts up or down
𝑔(𝑥) = √𝑥 − 1, ℎ(𝑥) = √𝑥 − 1
Reflection
𝑓(−𝑥)
𝑔(𝑥) = √−𝑥, ℎ(𝑥) = √−𝑥
Flips over x-or y-axis
−𝑓(𝑥)
𝑔(𝑥) = −√𝑥, ℎ(𝑥) = − √𝑥
In the x-axis
3
𝑔(𝑥) = √3𝑥, ℎ(𝑥) = √3𝑥
Horizontal Stretch or Shrink
Stretches away from or shrinks
3
𝑓(𝑎𝑥)
Shrink by a factor of
1
3
toward y-axis
3 1
1
𝑔(𝑥) = √ 𝑥, ℎ(𝑥) = √ 𝑥
2
2
Vertical Stretch or Shrink
𝑔(𝑥) = 4√𝑥, ℎ(𝑥) = 4 √𝑥
Stretch by a factor of 4
1
13
𝑔(𝑥) = √𝑥, ℎ(𝑥) = √𝑥
5
5
Shrink by a factor of 5
Stretches away from or shrinks
toward x-axis
3
𝑎 ∙ 𝑓(𝑥)
Stretch by a factor of 2
1