Radical Functions
Unit 2
Laws of Exponents
• X2
x is the base and 2 is the
exponent (the base can be a number or a
variable)
• Rule 1—Multiplying like bases
When multiplying like bases, keep the base
and ADD the exponents
Examples
3 3
2
5
4 4
2x
3
1
x x x
6
 4  4
3
2
p
Laws of Exponents
• Rule 2-- dividing like bases
When dividing like bases, keep the base and
subtract the exponents (top minus the
bottom)
Ex:
Examples
9
2

3
2
2
4x

11
8x
7
4
2 x  3x

5
24 x
Laws of Exponents
• Rule 3—Raising a power to another power
When raising a power to another power, you
keep the base and multiply the exponents
EX:
Examples
4 x 
2

 
 5x  
 3x   2 x 
 3 2x
3 3
2 5
3 2
4 4

Laws of Exponents
• Rule 4-- negative exponents
Do not leave any negative exponents in the
final answer, to make them positive move
from top to bottom or bottom to top (doesn’t
matter if the base is negative)
Ex:
Examples
3
2 
4m  3mb 
3 2
5
8

20
8
Laws of Exponents
• Rule 5--zero power
Anything raised to the zero power =‘s 1
Ex:
x x
4
4
3 5
4s t
3s t 24s 
2 5
1
Try these. Don’t leave any negative
exponents
3
4

3x y 2 xy
  3abc

3
9
a
bc

32
8
2
2



2
8

3a b 
4 3 2
( 2a ) 3
 2 x3x y
3
5
4

2 4
 5
4
Laws of Exponents
• Rule 6—rational exponents
We will come back to this rule after we discuss
radicals
Rational Exponents
• Exponents that are fractions.
•
𝑛
𝑚
𝑛
𝑥 𝑚 = 𝑥 for any positive integer n. The
exponent form indicates the principal
(positive) root.
1
2
• Generally, 𝑥 = 𝑥
Write in Radical Form
1. 27
2. 𝑦
1
3
2
5
3. 10,0000.75
4. 𝑚1.2
Write in Exponential Form
1.
7𝑥 3
2.
3
3.
3
4.
𝑎2
5𝑥𝑦
5
𝑦
4
6
Simplify
2
3
1. 𝑥 5 𝑦 6
2. 125
3. 32
4.
5
𝑥3
7
𝑥3
−
2
5
8
3
Simplify
5.
9
𝑦3
3
𝑦9
5
6.
7.
5∙ 5
6
4
3
4
Parts of a Radical
index
radical
𝑛
𝑚
radican
Radicals
• Principal root—positive root (for even
indexes)
• For a radical to be completely simplified, all
perfect nth
root factors should be removed
from underneath the radical, no fractions left
underneath the radical and no radicals left in
the denominator
• All even powered variables are perfect
squares—the sq. rt is ½ the power
Radicals are simplified if they have
• No factors that are perfect squares
• No radicals in the denominator
Rationalizing the denominator
• Multiply by a sq. rt. that will give you a perfect sq. in
the denominator so that you can eliminate the
radical
3
1.
7
2
54
3. 3
9x2
20 x 2
5. 5
16 x 4 y 6
100 x
2.
8
6
3
m
4. 4
8
Simplify, if possible
20
27
Simplify, if possible
28
108
Simplify, if possible
3
5
2
7
Simplify, if possible
5
8
6
12
Simplify, if possible
3 5
3
2 6
3
Simplifying Higher Order Radicals
•
3
54𝑥 5
•
3
162𝑥 5
Simplify
•
3
−27𝑥 7 𝑦 5 𝑧 6
•
5
−64𝑓 6 𝑔5 ℎ27
Examples
4
1. 64 x y
10
2. 8 x y
8
6
3. 20 x 7 y11z
4. 3 64 x 9 y15
5. 3 8 x 2 y 28
6. 4 32 x 4 y10 z 6
7. 5 96 x10 y 23
More Examples
1. 4 162 x 6 y 6
2. 3 16 x 9 y 6
3. 200 x 8 4 x 5
4. x 9 y16
5. 3  27 x 5 y 30
6. 4 8 x 4 y12 z
7. 5 5 x10 y 26