1. Radical Functions & Rational Exponents
Unit Objectives:
Simplify radical and rational exponent expressions
Solve radical equations
Find the inverse of a function: algebraically & graphically
Identify function attributes: domain and range
Perform function operations: add, subtract, multiply & compose
Today’s Objective:
I can simplify radical expressions.

2. Review of Exponent Properties
𝑏 𝑚 ⋅ 𝑏 𝑛 = 𝑏
𝑚+𝑛
(𝑏 𝑚 ) 𝑛 = 𝑏
𝑚⋅𝑛
(𝑎𝑏) 𝑛 =
𝑎 𝑛 ⋅ 𝑏 𝑛
𝑏 0 = 1
𝑏 𝑚 𝑏 𝑛 = 𝑏
1 𝑏 𝑛
𝑎 𝑏 𝑛 =
𝑎 𝑛 𝑏 𝑛
𝑚−𝑛
𝑏 −𝑛 =
Simplify with positive exponents only.
3 𝑥 −2 𝑦 3 𝑥 5 𝑦 7 −1
2 𝑥 5 ⋅3 𝑥 8
(3 𝑥 5 𝑦 −3 ) 2
9 𝑥 10 𝑦 6
6 𝑥 13
𝑥 7 𝑦 4 3

3. Roots & Radical Expressions
Powers
Roots
Radicals
2 2 = 4
2 is the square root of 4
4 =
2 2 = 2
2 3 = 8
2 is the cube root of 8
3 8 = = 2
2 4 = 16
2 is the fourth root of 16
4 16 = = 2
𝑎 𝑛 = 𝑏
a is the nth root of b
𝑛 𝑏 =
𝑛 𝑎 𝑛 = 𝑎
Index:
Degree of root
Radicand

4. Simplifying Radicals 𝑛 𝑎 𝑛 = 𝑎, if 𝑛 is odd 𝑎 , if 𝑛 is even 3 𝑥 12 =
Write radicand in factors raised to the nth power or less.
Take the nth root of all factors to the nth power.
Simplify in front of radical and under radical.
3 𝑥 12 =
3 𝑥 3 ⋅ 𝑥 3 ⋅ 𝑥 3 ⋅ 𝑥 3
25 =
5 2 = 5
3 125 = =
=𝑥⋅𝑥⋅𝑥⋅𝑥
= 𝑥 4 5
3 −27 =
3 (−3) 3 = −3
5 32 𝑥 10 =
⋅ 𝑥 5 ⋅ 𝑥 5
3 64 =
3 8⋅8
=2⋅𝑥⋅𝑥
= 2𝑥 2
= ⋅ 2 3 =
2⋅2= 4

5. Simplifying Radicals 𝑛 𝑎 𝑛 = 𝑎, if 𝑛 is odd 𝑎 , if 𝑛 is even
Write radicand in factors raised to the nth power or less.
Take the nth root of all factors to the nth power.
Simplify in front of radical and under radical.
3 54 𝑥 5 =
3 27⋅2⋅ 𝑥 3 ⋅ 𝑥 2
24 =
4⋅6
2 2 ⋅6 =
2 6
= ⋅2⋅ 𝑥 3 ⋅ 𝑥 2
=3𝑥 3 2 𝑥 2
3 24 =
3 8⋅3
4 48 𝑥 13 =
4 16⋅3⋅ 𝑥 4 ⋅ 𝑥 4 ⋅ 𝑥 4 ⋅𝑥
⋅3 =
2 3 3
= ⋅3⋅ 𝑥 4 ⋅ 𝑥 4 ⋅ 𝑥 4 ⋅𝑥
=2⋅𝑥⋅𝑥⋅𝑥 4 3𝑥
=2 |𝑥 3 | 4 3𝑥