7-5 Fractional Exponents Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

7-5 Fractional Exponents Warm Up Simplify each expression –3

7-5 Fractional Exponents 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. California Standards

7-5 Fractional Exponents index Vocabulary

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 2 3 = 2  2  2 = 8,

7-5 Fractional Exponents 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 b m = b n, then m = n. Divide both sides by 2. b 1 = b 2k

7-5 Fractional Exponents When b = 0, When b = 1, Helpful Hint

7-5 Fractional Exponents

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

7-5 Fractional Exponents Check It Out! Example 1 Simplify each expression. a. = 3 b. = = 15 b 1 n Use the definition of. b 1 n

7-5 Fractional Exponents 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

7-5 Fractional Exponents

7-5 Fractional Exponents Additional Example 2: Simplifying Expressions with Fractional Exponents Simplify each expression. A.B. Definition of = 243= 25

7-5 Fractional Exponents Check It Out! Example 2 Simplify each expression. a. = 8 b. = 1 Definition of = (1) 3

7-5 Fractional Exponents Check It Out! Example 2 Simplify each expression. = 81 Definition of c.

7-5 Fractional Exponents 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 m 2. Substitute 54 for s. Simplify inside the parentheses. Definition of The volume of the cube is 27 m 3.

7-5 Fractional Exponents 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. = 72  27 = 1944 The panda needs 1944 Calories per day to maintain health. Substitute 81 for m. Definition of

7-5 Fractional Exponents 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.

7-5 Fractional Exponents 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.

7-5 Fractional Exponents 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 Exponents Additional 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 Exponents Additional Example 4B: Properties of Exponents to Simplify Expressions Simplify. All variables represent nonnegative numbers. Power of a Product Property Product of Powers Property Simplify exponents.

7-5 Fractional Exponents Check It Out! Example 4a Simplify. All variables represent nonnegative numbers. Power of a Product Property Simplify exponents. Definition of

7-5 Fractional Exponents Check It Out! Example 4b Simplify. All variables represent nonnegative numbers. Power of a Product Property and Simplify. = xy

7-5 Fractional Exponents Lesson Quiz: Part I Simplify each expression

7-5 Fractional Exponents In an experiment, the approximate population P of a bacteria colony is given by, where t is the number of days since the start of the experiment. Find the population of the colony on the 8th day Simplify. All variables represent nonnegative numbers Lesson Quiz: Part II