Properties of Rational
Exponents
Section 7.2
WHAT YOU WILL LEARN:
1. Simplify expressions with rational exponents.
2. Use properties of rational exponents.
3. Write an expression involving rational exponents in simplest
form.
4. Perform operations with rational exponents.
5. Simplify expressions that have variables and rational exponents.
6. Write an expression involving variables and rational exponents in
simplest form.
7. Perform operations with rational exponents and variables.
Properties of Rational Exponents
Properties of Rational Exponents:
Property:
1. a m  a n  a m n
2. (am)n = amn
3. (ab)m = ambm
4.
a
m
1
 m ,a  0
a
Example:
1
2
3
2
1 3
(  )
2 2
3 3  3
3
2 2
(4 )  4
1
2
3
( 2 )
2
1
2
3 9
2
 4  64
3
1
2
(9  4)  9  4  3  2  6

25
1
2

1
1
2
25

1
5
Properties of Rational Exponents (cont.)
Properties of Rational Exponents:
Property:
Example:
m
5.
a
mn

a
,a  0
n
a
6
6
m
6.
a m a
( )  m ,a  0
b
b
5
2
1
2
6
1
3
5 1
(  )
2 2
1
3
 6  36
2
8
8
2
( )  1 
27
3
3
27
Using the Properties
• Simplify the expressions:
1.
1
2
1
4
1
2
1
3 2
5 5
2. (8  5 )
3.
(2  3 )
4
4

1
4
More Fun with Properties
4.
7
7
5.
(
1
3
12
4
1
3
1
3
)2
You Try
• Simplify:
1
2
1. 6  6
1
3
1
3
1
4 2
2. (27  6 )
3. (43  23 )
4.
6
6
2
3

 18
 1
5.  9 4
1
4





3

1
3
More Simplifying
• Simplify the expressions:
1.
3
2.
4
4  3 16
162
4
2
You Try
• Simplify:
1.
2.
3
25 3 5
3
32
3
4
Simplest Form - continued
• In order for a radical to be in simplest form,
you have to remove any perfect nth powers and
rationalize denominators. Example:
Write in simplest form:
1.
3
54
2.
5
3
4
You Try
• Write in simplest form:
1. 4
2. 4
64
7
8
Operations Using Radicals
• Two radicals expressions are “like radicals”
if they have the same index and the same
radicand. Example:
• Perform the indicated operation:
1.
1
5
1
5
7(6 )  2(6 )
2. 3
16  2
3
You Try
• Perform the indicated operation:
3
4
3
4
1.
5(4 )  3(4 )
2.
3
81  3
3
Simplifying Expressions Involving Variables
• Important!
n
n
n
x = x when n is odd.
x
n
= |x| when n is even.
Simplifying
• Simplify the expression. Assume all
variables are positive:
1
1.
3.
3
4
125 y 6
x4
8
y
2 10 2
(
9
u
v )
2.
4.
6 xy
1
3
1
2
2 x z 5
You Try
• Simplify the expression. Assume all variables
are positive.
1.
3
27z 9
1
2 2
2. (16g 4 h )
3.
4
x5
y10
4. 18rs
2
3
1
4 3
6r t
Writing Variable Expressions in Simplest Form
• Write the expression in simplest form.
Assume all variables are positive.
1.
5
5 9 13
5a b c
2.
3
x
y7
You Try
• Write the expression in simplest form.
Assume all variables are positive.
1.
2.
4
5
12d 4 e 9 f 14
g2
h7
Adding and Subtracting Expressions Involving Variables
• Perform the indicated operation. Assume
all variables are positive. 1
1
1. 5 y  6 y
3. 33 5x5  x3 40x2
2. 2 xy 3  7 xy 3
You Try
• Perform the indicated operations. Assume all
variables are positive.
1. 8 x  3 x
1
4
2. 3gh  6 gh
1
4
3. 24 6 x 5  x 4 6 x
Homework
page 411, 22-30 even, 34-62 even, 66-72
even, 76, 80
: