Name _______________________________________ Date __________________ Class __________________
Section 8.6
Page 1
Radical Expressions and Rational Exponents
Use Properties of nth Roots to simplify radical expressions.
n
Product Property:
Simplify:
ab  n a  n b
4
81x 8 .
4
34  x 4  x 4
4
34
4
x4
4
Factor into perfect fourth roots.
x4
Use the Product Property.
3xx
Think:
2
3x
Quotient Property:
Simplify:
n
3
3
2
2

3
3
x 3 3 22
3
4
x4  x .
Simplify the numerator. Think:
x3
3
3 4  3 and
Use the Quotient Property.
2
x3
3
4
Always rationalize the denominator
when an expression contains a radical
in the denominator.
x9
3
a n  a , so
a na

b nb
x9
.
2
3
n
23
x3 3 4
2
2
2

3
2
3
2
x9  3 x3
3
x3
3
x 3  x  x  x.
Rationalize the denominator.
Use the Product Property.
Simplify.
Simplify each expression.
1.
3
x8  3 x 4
2.
________________________
4.
5
64
2x 10
________________________
4
x8
6
3.
________________________
5.
3
2x  3 4 x 2
________________________
3
125x 6
________________________
6.
4
625x 8
________________________
Name _______________________________________ Date __________________ Class __________________
Section 8.6
Page 2
Radical Expressions and Rational Exponents (continued)
The n th root of a number can be represented using a rational, or fractional,
1
a  an .
n
exponent:
This means to take the nth root of a.
1
Examples: 1212  121  11
1
216 3  3 216  6
1
256 4  4 256  4
m
Powers and roots can be expressed using rational exponents a n 
2

4
 32 5
3
64


32

5
n
m
.
The denominator is the root and
the numerator is the power.
Examples:
64 3 
 a
2
 42  16

4
  2  16
4
To write expressions using rational exponents, use the definitions.
n
1
a  a n and
n
m
am  a n
1
5  52
Examples:
Note that
4
3
 a
n
m
 n am .
63  6 4
Think: The root is n  4.
The power is m  3.
Write each expression in radical form and simplify.
4
7. 27 3 

3
27

4
3
3
8. 49 2
9. 16 4
________________________
________________________
________________________
Write each expression by using rational exponents.
10.
5
42 Think: m  2, n  5
11.
________________________
19
12.
________________________
4
65
________________________
Simplify each expression.
1
 24  3
13.  3 
 3x 
________________________
14.
49  3 8x 6
________________________
15.
117
13
________________________
Section 8.6
Page 3
Radical Expressions and Rational Exponents
Simplify each expression. Assume all variables are positive.
1.
4
 2x 
8
 3  2x 
6
2.
________________________
x 16 y 8
81
4
3.
________________________
3
x7
27 x 3
________________________
Write each expression in radical form, and simplify.
2
4. 216 3
5. 1000
________________________

2
3

6. 16x 3
________________________
3
2
________________________
Write each expression by using rational exponents.
7.
5
 3x 
4
8.
________________________

5
6

3
9.
________________________
4
30x 3
________________________
Simplify each expression.
3
10.
1
25 4
7
 25 4
11.  64 
________________________
________________________
1
 x3 3
13. 

 125 
14.
________________________
16.

3
8x
9

2
________________________
 x8  4
12.  4 
y 
1
3
 8x 
2
18 3
 
3
y6
________________________
17.  3 x 
2
3
3x 
7
3
________________________
________________________

15. a 4 b8

1
4
________________________
 m8 
18.  12 
n 
1
4
________________________
Page 1
1. x4
3.
3
2.
x 2 4 216
6
125  3 x 6
 5x2
4.
2
x2
5. 2x
6. 5x2
Page 2
7. 81
8.

49

3
 343
9.

4
16
2
1
10. 4 5
11. 19 2
5

3
8
2
x
12. 6 4
13.
14. 14x2
15. 3
Page 3
1.16x4
2.
x4y 2
3
3.
x3 x
3
4. 36
5.
1
100
6. 64x4
7.
4
5
(3 x )
1
8.
3
9. 30 4  x 4
11. 4
3
( 6)5
10.
1
125
12.
x5
y3
13.
x
5
14. 4x12y2
15.
1
ab 2
16. 4x6
17. 27x3
18.
n3
m2
x