Exponents 6 2 Rational Exponents Warm Up Lesson

Exponents 6 -2 Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1 Holt Mc. Dougal

6 -2 Rational Exponents Warm Up Simplify each expression. 1. 6 2. 3. 0 4 4. 1 10 5. 6. – 3 Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Objective Evaluate and simplify expressions containing rational exponents. Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Vocabulary index Holt Mc. Dougal Algebra 1

6 -2 Rational 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 = 2 2 2 = 8, Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Another way to write nth roots is by using fractional exponents. For example, for b >1, suppose b 1 = b 2 k 1 = 2 k Square both sides. Power of a Power Property If bm = bn, then m = n. Divide both sides by 2. So for all b > 1, Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Helpful Hint When b = 0, When b = 1, Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Additional Example 1: Simplifying b Simplify each expression. 1 n A. Use the definition of 1 b n. =7 B. Use the definition of =2+3=5 Holt Mc. Dougal Algebra 1 1 b n.

6 -2 Rational Exponents Check It Out! Example 1 Simplify each expression. a. Use the definition 1 of b n. =3 b. Use the definition of = 11 + 4 = 15 Holt Mc. Dougal Algebra 1 1 b n.

6 -2 Rational 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 Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Additional Example 2: Simplifying Expressions with Fractional Exponents Simplify each expression. A. B. Definition of = 243 Holt Mc. Dougal Algebra 1 = 25

6 -2 Rational Exponents Check It Out! Example 2 Simplify each expression. a. b. Definition of = (1)3 =8 Holt Mc. Dougal Algebra 1 =1

6 -2 Rational Exponents Check It Out! Example 2 Simplify each expression. c. Definition of = 81 Holt Mc. Dougal Algebra 1

6 -2 Rational 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. Holt Mc. Dougal Algebra 1

6 -2 Rational 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. Substitute 81 for m. Definition of = 72 27 = 1944 Holt Mc. Dougal Algebra 1 The panda needs 1944 Calories per day to maintain health.

6 -2 Rational 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. Holt Mc. Dougal Algebra 1

6 -2 Rational 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. Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Helpful Hint When you are told that all variables represent nonnegative numbers, you do not need to use absolute values in your answer. Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Additional Example 4 A: 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. Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Additional Example 4 B: Properties of Exponents to Simplify Expressions Simplify. All variables represent nonnegative numbers. • • Power of a Product Property Simplify exponents. Product of Powers Property Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Check It Out! Example 4 a Simplify. All variables represent nonnegative numbers. Definition of Power of a Product Property Simplify exponents. Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Check It Out! Example 4 a Simplify. All variables represent nonnegative numbers. Definition of Power of a Product Property Simplify exponents. Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Check It Out! Example 4 b Simplify. All variables represent nonnegative numbers. Power of a Product Property and Simplify. = xy Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Lesson Quiz: Part I Simplify each expression. 1. 2 2. 9 3. 729 4. 128 Holt Mc. Dougal Algebra 1

6 -2 Rational Exponents Lesson Quiz: Part II 5. In an experiment, the approximate population P of a bacteria colony is given by , where t is the number of days since start of the experiment. Find the population of the colony on the 8 th day. 480 Simplify. All variables represent nonnegative numbers. 6. 7. Holt Mc. Dougal Algebra 1