7-5 Fractional Exponents Recall that the radical symbol is used to indicate roots. The index is the small number to the left of the radical symbol that

7-5 Fractional Exponents

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 = 222 = 8,


Another way to write nth roots is by using fractional exponents. For example, for b >1, suppose

1 = 2k

So for all b > 1,

Square both sides.

Power of a Power Property

If bm = bn, then m = n.

Divide both sides by 2.

b1 = b2k


When b = 0,

When b = 1,

Helpful Hint


7-5 Fractional ExponentsAdditional Example 1: Simplifying b

1n

Simplify each expression.

A.

= 7

b1nUse the definition of .

B.


= 2 + 3 = 5

7-5 Fractional ExponentsCheck It Out! Example 1


a.

= 3

b.

= 11 + 4

= 15




Check up: p. 425 #’s 2,5, 10, 13


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 PowerProperty

Definition of


7-5 Fractional ExponentsAdditional Example 2: Simplifying Expressions with

Fractional Exponents


A. B.

Definition of

= 243 = 25



a.

= 8

b.

= 1

Definition of

= (1)3



= 81

Definition of

c.


Check up: p. 425 #’s 14, 16, 19

7-5 Fractional ExponentsAdditional 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.


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.

= 7227 = 1944

The panda needs 1944 Calories per day to maintain health.

Substitute 81 for m.

Definition of


Check up: p. 425 # 22


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.


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.


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

Helpful Hint

7-5 Fractional ExponentsAdditional Example 4A: Properties of Exponents to

Simplify Expressions

Simplify. All variables represent nonnegative numbers.

Power of a Product Property

Power of a Power Property

Simplify exponents.

•

Definition of

7-5 Fractional ExponentsAdditional Example 4B: Properties of Exponents to

Simplify Expressions



Product of Powers Property

Simplify exponents.

• •

7-5 Fractional ExponentsCheck It Out! Example 4a



Simplify exponents.

Definition of

7-5 Fractional ExponentsCheck It Out! Example 4b


Power of a Product Property and

Simplify.

= xy


Check up: p. 425 #’s 23,28,29

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7-5 Fractional Exponents Recall that the radical symbol is used to indicate roots. The index is the small number to the left of the radical symbol that