NAME
DATE
6-6
PERIOD
Study Guide and Intervention
Rational Exponents
Rational Exponents and Radicals
1
Definition of b −n
m
Definition of b −n
Example 1
For any real number b and any positive integer n,
1
n
b −n = √
b , except when b < 0 and n is even.
For any nonzero real number b, and any integers m and n, with n > 1,
m
n
n
b −n = √
bm =( √
b ) , except when b < 0 and n is even.
m
1
−
( -125 )
Notice that 28 > 0.
1
−
-8 3
.
Evaluate −
Example 2
Write 28 2 in radical form.
Notice that -8 < 0, -125 < 0, and 3 is odd.
1
−
28 = √28
2
( -125 )
-8
−
= √
22 7
= √
22 √7
= 2 √
7
1
−
3
3
√
-8
√
-125
-2
=−
-5
2
=−
5
=−
3
Exercises
Write each expression in radical form, or write each radical in exponential form.
1
−
3
−
2. 15 3
7
√
11
4. √47
3. 300 2
3
√
15
3
5 2
5. √3a
b
5
−
1
−
1
−
3000 √
3
6.
2
−
5
−
1
−
33 a3 b3
47 2
4
162p5
√
3 24 p4
Evaluate each expression.
2
−
8. 216 3
9
6
Chapter 6
1
−
1
−
7. -27 3
9. (0.0004) 2
0.02
38
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
−
1. 11 7
NAME
DATE
6-6
PERIOD
Study Guide and Intervention
(continued)
Simplify Expressions All the properties of powers from Lesson 6-1 apply to rational
exponents. When you simplify expressions with rational exponents, leave the exponent in
rational form, and write the expression with all positive exponents. Any exponents in the
denominator must be positive integers.
When you simplify radical expressions, you may use rational exponents to simplify, but your
answer should be in radical form. Use the smallest index possible.
2
−
Example 1
3
−
2
−
3
2
−
+−
y3 y8 = y3
8
3
−
Example 2
Simplify y 3 y 8 .
4
Simplify √
144x6 .
1
25
−
4
−
√
144x6 = (144x6) 4
= y 24
= (24 32
1
−
x6) 4
1
−
1
−
1
−
= (24) 4 (32) 4 (x6) 4
1
−
1
−
3
−
= 2 3 2 x 2 = 2x (3x) 2 = 2x √
3x
Exercises
Simplify each expression.
2. (y 3 ) 4
x2
y2
6
−
4
−
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
2 −
−
1. x 5 x 5
2
6 −
−
12
−
m 25
3
−
p2
4
1 −
−
6. (s 6 ) 3
4
−
5. x 8 x 3
2
−
41
−
x 24
s9
1
−
x2
8. −
1
p
7. −1
−
6
9. √
128
−
p3
p
3
−
1
−
4. (m5 ) 5
7
4
−
−
3. p 5 ․ p 10
x3
2
−
1
5
−
x6
−
or −
1
x
−
x6
3
4
10. √49
√
7
5
11. √
288
6
2 √
2
12. √
32 3 √
16
5
2 √
9
48 √
2
3
3
√
13. √25
125
6
25 √
5
Chapter 6
a √
b4
√
ab3
6
14. √
16
15. −
6
√a
b5
√
−
3
√
4
b
39
Glencoe Algebra 2
Lesson 6-6
Rational Exponents