1. Recall that the radical symbol is used to indicate roots
Recall that the radical symbol is used to indicate roots. The index is the small number to the left of the radical symbol that tells which root to take. For example represents a cubic root. Since 23 = 222 = 8,

2. Another way to write nth roots is by using fractional exponents
Another way to write nth roots is by using fractional exponents. For example, for b >1, suppose
Square both sides.
b1 = b2k
Power of a Power Property
1 = 2k
If bm = bn, then m = n.
Divide both sides by 2.
So for all b > 1,

3. When b = 0,
When b = 1,
Helpful Hint

5. Additional Example 1: Simplifying b
Simplify each expression. A. b 1 n
Use the definition of
= 7 B. b 1 n
Use the definition of
= = 5

6. Simplify each expression.
Check It Out! Example 1
Simplify each expression. a. b 1 n
Use the definition of
= 3 b. b 1 n
Use the definition of =
= 15

7. Check up: p. 425 #’s 2,5, 10, 13

8. A fractional exponent can have a numerator other than 1, as in the expression . You can write the exponent as a product in two different ways.
Power of a Power
Property
Definition of

10. Additional Example 2: Simplifying Expressions with Fractional Exponents
Simplify each expression. A. B.
Definition of
= 243
= 25

11. Check It Out! Example 2
Simplify each expression. a. b.
Definition of
= (1)3
= 8
= 1

12. Check It Out! Example 2
Simplify each expression. c.
Definition of
= 81

13. Check up: p. 425 #’s 14, 16, 19

14. Additional Example 3: Application
Given a cube with surface area S, the volume V of the cube can be found by using the formula
Find the volume of a cube with surface area 54 m2.
Substitute 54 for s.
Simplify inside the parentheses.
Definition of
The volume of the cube is 27 m3.

15. The approximate number of Calories C that an
Check It Out! Example 3
The approximate number of Calories C that an 2
animal needs each day is given by , where m is the animal’s mass in kilograms. Find the number of Calories that an 81 kg panda needs each day.
Substitute 81 for m.
Definition of
The panda needs 1944 Calories per day to maintain health.
= 7227 = 1944

16. Check up: p. 425 # 22

17. Remember that always indicates a nonnegative square root
Remember that always indicates a nonnegative square root. When you simplify variable expressions that contain , such as , the answer cannot be negative. But x may be negative. Therefore you simplify as |x| to ensure the answer is nonnegative.

18. When n is even, you must simplify to |x|, because you do not know whether x is positive or negative. When n is odd, simplify to x.

19. When you are told that all variables represent nonnegative numbers, you do not need to use absolute values in your answer.
Helpful Hint

20. Additional Example 4A: Properties of Exponents to Simplify Expressions
Simplify. All variables represent nonnegative numbers.
Definition of
Power of a Product Property •
Power of a Power Property
Simplify exponents.

21. Additional Example 4B: Properties of Exponents to Simplify Expressions
Simplify. All variables represent nonnegative numbers. •
Power of a Product Property
Simplify exponents.
Product of Powers Property

22. Check It Out! Example 4a
Simplify. All variables represent nonnegative numbers.
Definition of
Power of a Product Property
Simplify exponents.

23. Check It Out! Example 4b
Simplify. All variables represent nonnegative numbers.
Power of a Product Property and
Simplify.
= xy

24. Check up: p. 425 #’s 23,28,29