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Chapter 6 38 Glencoe Algebra 2

Rational Exponents and Radicals

Write 28 1 −

2 in radical form.

Notice that 28 > 0.

28 1 − 2 = √ �� 28

= √ �� 22 � 7

= √ � 22 � √ � 7

= 2 √ � 7

Evaluate (

-8 −

-125 )

1 −

3 .

Notice that -8 < 0, -125 < 0, and 3 is odd.

( -8 − -125

) 1 − 3 = 3 √

�� -8 −

3 √ ��� -125

= -2 − -5

= 2 − 5

Exercises

Write each expression in radical form, or write each radical in exponential form.

1. 11 1 − 7 2. 15

1 − 3 3. 300

3 − 2

4. √ �� 47 5. 3 √

��� 3a5b2 6. 4

√ ��� 162p5

Evaluate each expression.

7. -27 2 − 3 8. 216

1 − 3 9. (0.0004)

1 − 2

Definition of b 1 − n

For any real number b and any positive integer n,

b 1 − n = n √

� b , except when b < 0 and n is even.

Definition of b m − n

For any nonzero real number b, and any integers m and n, with n > 1,

b m − n = n √

� bm = (

n

√

�

b ) m

, except when b < 0 and n is even.

Example 1 Example 2

6-6 Study Guide and InterventionRational Exponents

7 √

�

11 3 √

��

15 3000 √

�

3

47 1 −

2 3

1 −

3 a

5 −

3 b

2 −

3 3 � 2

1 −

4 � p

5 −

4

9 6 0.02

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Chapter 6 39 Glencoe Algebra 2

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.

Simplify y 2 −

3 � y

3 −

8 .

y 2 − 3 � y

3 − 8 = y

2 − 3 + 3 −

8 = y

25 − 24

Simplify 4 √

���

144x6 .

4 √

��� 144x6 = (144x6) 1 − 4

= (24 � 32 � x6) 1 − 4

= (24) 1 − 4

� (32) 1 − 4 � (x6)

1 − 4

= 2 � 3 1 − 2 � x

3 − 2 = 2x � (3x)

1 − 2 = 2x √ � 3x

ExercisesSimplify each expression.

1. x 4 − 5 � x

6 − 5 2. ( y

2 − 3 )

3 − 4 3. p

4 − 5 ․ p

7 − 10

4. ( m 6 − 5 )

2 − 5 5. x

3 − 8 � x

4 − 3 6. ( s

1 − 6 )

4 − 3

7. p −

p 1 − 3 8. x

1 − 2 −

x 1 − 3 9. 6 √

�� 128

10. 4 √ �� 49 11. 5 √

�� 288 12. √ �� 32 � 3 √ �� 16

13. 3 √ �� 25 � √ �� 125 14. 6 √

�� 16 15. a

3 √

� b4 −

√ �� ab3

6-6 Study Guide and Intervention (continued)

Rational Exponents

Example 1 Example 2

x2 y 1 −

2 p

3 −

2

m 12

−

25 x

41 −

24 s

2 −

9

p 2 −

3 1

−

x 1 −

6 or x

5 −

6 −

x 2

6 √

�

2

√ �

7 2 5 √

�

9 48 √

�

2

25 6 √ �

5 3 √

�

4 √

� a

6 √

�

b5 −

b

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