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Name
Class
Date
Notes
6-4
Rational Exponents
You can simplify a number with a rational exponent by converting the expression to a
radical expression:
1
1
1
x n  n x , for n  0
92  2 9  3
83  3 8  2
You can simplify the product of numbers with rational exponents m and n by raising
the number to the sum of the exponents using the rule
a m  a n  a m n
What is the simplified form of each expression?
1
1
a. 36 4  36 4
1
11
4
1
Use a m  a n  a m  n .
36 4  36 4  36 4
1
 36 2
Add.
 2 36
Use x n  n x .
6
Simplify.
2
3
1
3
4
b. Write (6 x )(2 x ) in simplified form.
2
3
2
3
(6 x 3 )(2 x 4 )  6  2  x 3  x 4
Commutative and Associative
Properties of Multiplication
2
3
 6  2  x3  x4
 12x
Use x  x  x
m
17
12
n
m+n
.
Simplify.
Exercises
Simplify each expression. Assume that all variables are positive.
1
1
1
1
1
1
2
1. 53  53
2. (2 y 4 )(3 y 3 )
3. (11) 3  (11) 3  (11) 3
2
1
4.  y 3 y 5
1
1
5. 54  54
1
2
6. ( 3 x 6 )(7 x 6 )
Name
Class
Date
Notes (continued)
6-4
Rational Exponents
To write an expression with rational exponents in simplest form, simplify all exponents and
write every exponent as a positive number using the following rules for a ≠ 0 and rational
numbers m and n:
1
a n  n
a
What is (8 x 9 y 3 )
 23
1
 am

m
a
(am )n  amn
(ab)m  ambm
in simplest form?
2
2
(8 x9 y 3 ) 3  (23 x9 y 3 ) 3
Factor any numerical coefficients.
2
2
2
 (23 ) 3 ( x9 ) 3 ( y 3 ) 3 Use the property (ab)m  ambm
 22 x6 y 2
m n
Multiply exponents, using the property (a )  a

y2
22 x 6
Write every exponent as a positive number.

y2
4 x6
Simplify.
Exercises
Write each expression in simplest form. Assume that all variables
are positive.


1
8 2

1
2 2
7. 16x 2 y
10. 25x
6
y

 
8. z 3

1
9
1

4
9.  2x 4 


1

11. 8a 3b9

2
3
 16 z 4  2

12. 
 25 x8 


mn .