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Chapter 5 Section 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Product Rule and Power Rules for Exponents Use exponents. Use the product rule for exponents. Use the rule ( a m ) n = a mn. Use the rule ( ab ) m = a m b n. Use the rule Use combinations of rules. Use the rules for exponents in a geometric application. 1 1 4 4 3 3 2 2 6 6 5 5 5.15.1 7 7

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Slide 5.1 - 3 Use exponents.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use exponents. Recall from Section 1.2 that in the expression 5 2, the number 5 is the base and 2 is the exponent or power. The expression 5 2 is called an exponential expression. Although we do not usually write the exponent when it is 1, in general, for any quantity a, a 1 = a. Slide 5.1 - 4

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Write 2 · 2 · 2 in exponential form and evaluate. Solution: Using Exponents Slide 5.1 - 5

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate. Name the base and the exponent. EXAMPLE 2 Evaluating Exponential Expressions Slide 5.1 - 6 Solution: Base:Exponent: BaseExponent Note the difference between these two examples. The absence of parentheses in the first part indicate that the exponent applies only to the base 2, not −2.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Slide 5.1 - 7 Use the product rule for exponents.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use the product rule for exponents. By the definition of exponents, Generalizing from this example suggests the product rule for exponents. Slide 5.1 - 8 For any positive integers m and n, a m · a n = a m + n. (Keep the same base; add the exponents.) Example: 6 2 · 6 5 = 6 7 Do not multiply the bases when using the product rule. Keep the same base and add the exponents. For example 6 2 · 6 5 = 6 7, not 36 7.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Using the Product Rule Slide 5.1 - 9 Use the product rule for exponents to find each product if possible. a) b) c) d) e) f) The product rule does not apply. Be sure you understand the difference between adding and multiplying exponential expressions. For example, but

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Slide 5.1 - 10 Use the rule (a m ) n = a mn.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley We can simplify an expression such as (8 3 ) 2 with the product rule for exponents. Use the rule (a m ) n = a mn. Slide 5.1 - 11 The exponents in (8 3 ) 2 are multiplied to give the exponent in 8 6. This example suggests power rule (a) for exponents. For any positive number integers m and n, (a m ) n = a mn. (Raise a power to a power by multiplying exponents.) Example:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Using Power Rule (a) Slide 5.1 - 12 Simplify. Be careful not to confuse the product rule, where 4 2 · 4 3 = 4 2+3 = 4 5 =1024 with the power rule (a) where (4 2 ) 3 = 4 2 · 3 = 4 6 = 4096.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Use the rule (ab) m = a m b m. Slide 5.1 - 13

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use the rule (ab) m = a m b m. We can rewrite the expression (4x) 3 as follows. Slide 5.1 - 14 This example suggests power rule (b) for exponents. For any positive integer m, (ab) m = a m b m. (Raise a product to a power by raising each factor to the power.) Example:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Simplify. Solution: Using Power Rule (b) Slide 5.1 - 15 Power rule (b) does not apply to a sum. For example,, but Use power rule (b) only if there is one term inside parentheses.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 Objective 5 Use the rule Slide 5.1 - 16

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use the rule Slide 5.1 - 17 Since the quotient can be written aswe use this fact and power rule (b) to get power rule (c) for exponents. For any positive integer m, (Raise a quotient to a power by raising both numerator and denominator to the power.) Example:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Simplify. Solution: Using Power Rule (c) Slide 5.1 - 18 In general, 1 n = 1, for any integer n.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The rules for exponents discussed in this section are summarized in the box. Slide 5.1 - 19 Rules of Exponents These rules are basic to the study of algebra and should be memorized.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6 Objective 6 Use combinations of rules. Slide 5.1 - 20

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Slide 5.1 - 21 Simplify Solution: Use Combinations of Rules

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7 Objective 7 Use the rules for exponents in a geometric application. Slide 5.1 - 22

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Using Area Formulas Slide 5.1 - 23 Find an expression that represents the area of the figure. Solution:

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