1. Chapter 5 Section 1

2. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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. 5.1 2 3 4 5 6 7

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 1 Use exponents. Slide 5.1-3

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write 2 · 2 · 2 in exponential form and evaluate. Solution: Slide 5.1-5 EXAMPLE 1 Using Exponents

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluate. Name the base and the exponent. 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. Slide 5.1-6 EXAMPLE 2 Evaluating Exponential Expressions 6 2 

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Use the product rule for exponents. Slide 5.1-7

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the product rule for exponents. By the definition of exponents, Product Rule for Exponents 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 Generalizing from this example suggests the product rule for exponents. 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. Slide 5.1-8

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Use the product rule for exponents to find each product if possible. The product rule does not apply. Be sure you understand the difference between adding and multiplying exponential expressions. For example, Slide 5.1-9 EXAMPLE 3 Using the Product Rule

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 3 Use the rule (a m ) n = a mn. Slide 5.1-10

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. We can simplify an expression such as (8 3 ) 2 with the product rule for exponents. The exponents in (8 3 ) 2 are multiplied to give the exponent in 8 6. 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: Slide 5.1-11 Use the rule (a m ) n = a mn.

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: 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. Slide 5.1-12 EXAMPLE 4 Using Power Rule (a)

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 4 Use the rule (ab) m = a m b m. Slide 5.1-13

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the rule (ab) m = a m b m. We can rewrite the expression (4x) 3 as follows. 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: Slide 5.1-14

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify. Solution: Power rule (b) does not apply to a sum. For example,,but Use power rule (b) only if there is one term inside parentheses. Slide 5.1-15 EXAMPLE 5 Using Power Rule (b)

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 5 Use the rule. Slide 5.1-16

17. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the rule Since the quotient can be written as we use this fact and power rule (b) to get power rule (c) for exponents. 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: Slide 5.1-17

18. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify. Solution: In general, 1 n = 1, for any integer n. Slide 5.1-18 EXAMPLE 6 Using Power Rule (c)

19. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The rules for exponents discussed in this section are summarized in the box. Rules of Exponents These rules are basic to the study of algebra and should be memorized. Slide 5.1-19

20. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 6 Use combinations of rules. Slide 5.1-20

21. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify. Solution : Slide 5.1-21 EXAMPLE 7 Using Combinations of Rules

22. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 7 Use the rules for exponents in a geometry application. Slide 5.1-22

23. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write an expression that represents the area of the figure. Assume x>0. Solution: Slide 5.1-23 EXAMPLE 8 Using Area Formulas

24. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Homework Pg 301-302 # 16-90 Even