Warm Up Simplify each expression. 1. 2. 3. 4. 5. 6. 6 4 1 10 –3

Objective Evaluate and simplify expressions containing rational exponents.

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,

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. Power of a Power Property b1 = b2k 1 = 2k If bm = bn, then m = n. Divide both sides by 2. So for all b > 1,

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

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 . = 2 + 3 = 5

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 . = 11 + 4 = 15

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

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

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

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

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.

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

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.

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

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.

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

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

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

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