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.
Review of Exponent Properties
𝑏𝑚 ⋅ 𝑏𝑛 = 𝑏
𝑏𝑚
𝑚−𝑛
=
𝑏
𝑏𝑛
𝑚+𝑛
(𝑏 𝑚 )𝑛 = 𝑏 𝑚⋅𝑛
𝑏 −𝑛
1
= 𝑛
𝑏
Simplify with positive exponents only.
2𝑥 5 ⋅ 3𝑥 8
(3𝑥 5 𝑦 −3 )2
6𝑥 13
9𝑥 10
𝑦6
(𝑎𝑏)𝑛 = 𝑎𝑛 ⋅ 𝑏 𝑛
𝑎
𝑏
𝑛
𝑎𝑛
= 𝑛
𝑏
𝑏0 = 1
−1
−2
3
3𝑥 𝑦
𝑥5𝑦7
𝑥7𝑦4
3
Roots & Radical Expressions
Powers
Roots
Radicals
22 = 4
2 is the square root of 4
23 = 8
2 is the cube root of 8
24 = 16
2 is the fourth root of 16
𝑎𝑛 =𝑏
a is the nth root of b
Index:
Degree of
root
n
b
4 = 22 = 2
3
4
𝑛
Radicand
3
8 = 23 = 2
4
16 = 24 = 2
𝑏 = 𝑛 𝑎𝑛 = 𝑎
Simplifying Radicals
𝑛
𝑎𝑛
𝑎, if 𝑛 is odd 1. Write radicand in factors raised to the nth
power or less.
𝑎 , if 𝑛 is even
=
2. Take the nth root of all factors to the nth power.
3. Simplify in front of radical and under radical.
52
25 =
3
=5
3
𝑥 12
3
= 𝑥3 ⋅ 𝑥3 ⋅ 𝑥3 ⋅ 𝑥3
= 𝑥 ⋅ 𝑥 ⋅ 𝑥 ⋅ 𝑥 = 𝑥4
125 = 3 53 =5
3
3
−27 = 3 (−3)3 = −3
64 =
=
3
3
8⋅8
23 ⋅ 23 = 2 ⋅ 2 = 4
5
32𝑥 10
=
5
25 ⋅ 𝑥 5 ⋅ 𝑥 5
= 2 ⋅ 𝑥 ⋅ 𝑥 = 2𝑥 2
Simplifying Radicals
𝑛
𝑎𝑛
𝑎, if 𝑛 is odd 1. Write radicand in factors raised to the nth
power or less.
𝑎 , if 𝑛 is even
=
2. Take the nth root of all factors to the nth power.
3. Simplify in front of radical and under radical.
3
24 =
3
4⋅6
22 ⋅ 6 = 2 6
3
24 =
3
=
4
8⋅3
54𝑥 5 =
48𝑥 13 =
3
23 ⋅ 3 = 2 3
=
4
3
27 ⋅ 2 ⋅ 𝑥 3 ⋅ 𝑥 2
3
33
4
⋅2⋅
𝑥3
⋅
𝑥2
3
= 3𝑥 2𝑥 2
16 ⋅ 3 ⋅ 𝑥 4 ⋅ 𝑥 4 ⋅ 𝑥 4 ⋅ 𝑥
24 ⋅ 3 ⋅ 𝑥 4 ⋅ 𝑥 4 ⋅ 𝑥 4 ⋅ 𝑥
4
= 2 ⋅ 𝑥 ⋅ 𝑥 ⋅ 𝑥 3𝑥
=
4
3
2|𝑥 |
3𝑥