Algebra II/Trig Honors
Unit 3 Day 2: Apply Properties of Rational Exponents
Objective: Simplify expression involving rational exponents using properties of exponents
The properties of integer (positive/negative whole number) exponents you learned in the previous
chapter apply to rational exponents as well.
Properties of Rational Exponents
Property
Example
m
n
1. a  a 
1. 51 2  5 3 2 
 
2. a
m n
 

2. 35 2
3. ab  
5.
m

am

an
12
a0
4. 36 1 2 
a0
5.
b0
 27 
6.  
 64 
m
a
6.   
b

3. 16  9 
m
4. a
2

45 2

41 2
13

Example 1: Use properties of exponents
Use the properties of rational exponents to simplify the expression.
a. 71 4  71 2
d.
5
51 3

b. 61 2  41 3
 13
 42
e.  1
 63






2

2

c. 4 5  35

1 5
Example 2: Apply properties of exponents
2
A mammal’s surface area S (in square centimeters) can be approximated by the model S  km 3 where
m is the mass (in grams) of the mammal and k is the constant. The values of k for some mammals are
shown below. Approximate the surface area of a rabbit that has a mass of 3.4 kilograms ( 3.4  10 3
grams).
Mammal
k
Sheep
8.4
Rabbit
9.75
Horse
10.0
Human
11.0
Monkey
11.8
Bat
57.5
Properties 3 and 6 from the previous page can be expressed using radical notation by replacing m with
1
for some integer n greater than 1.
n
Exponent Property
Radical Property
3. ab  
m
n
m
a

b
b0
a
6.   
b
a b 
n
b0
Example 3: Use properties of radicals
Use the properties of radicals to simplify the expression.
a.
3
12  3 18
4
b.
4
80
5

Radicals in Simplest Form - ______________________________________________________
_____________________________________________________________________________
o When simplifying a fraction, we will need to get a perfect nth power in the denominator
to rationalize.
Example 4: Write radicals in simplest form
Write the expression in simplest form.
a. 3 135
5
7
5
8
b.
Note: Make the
denominator a perfect 5th
power. Keep the numbers
as small as possible.

Like Radicals - ____________________________________________________________
n
a
Example 5: Add and subtract like radicals and roots
Simplify the expression.
a.
4
10  74 10
 
 
b. 2 81 5  10 81 5
c.
3
54  3 2
When working with variable expressions, we may sometimes need absolute values when simplifying
with radicals or rational exponents. This is because a variable can be positive, negative, or zero, and we
do not know which unless the directions say “Assume all variables are positive.”
When n is odd
n
xn  x
7
n
When n is even
xn  x
57 
7
Ex.
 57
4
Ex.

34 
 34
4

Example 6: Simplify expressions involving variables
Simplify the expression. Assume all variables are positive. (This means you will not need absolute
values even for even roots).

a.
3
64 y 6
b. 27 p 3 q12
c.
4
m4
n8
d.

13
14 xy1 3
2 x 3 4 z 6
Example 7: Write variable expressions in simplest form
Write the expression in simplest form. Assume all variables are positive.
a.
5
4a 8b14c 5
b.
3
x
y8
Example 8: Add and subtract expressions involving variables
Perform the indicated operation. Assume all variables are positive.
a.
1
3
w
w
5
5
c. 123 2 z 5  z 3 54 z 2
HW: Page 176 #4-68 (M4), 70, 72, 78, 82ab
b. 3xy1 4  8 xy1 4