4.1 The Product Rule and Power Rules for Exponents

1. 4.1 The Product Rule and Power Rules for Exponents

2. Objective 1 Use exponents. Slide 4.1-3

3. Use exponents. Recall from Section 1.2 that in the expression 52, the number 5 is the baseand 2 is theexponentor power.The expression52is called an exponential expression.Although we do not usually write the exponent when it is 1, in general, for any quantity a, a1= a. Slide 4.1-4

4. CLASSROOM EXAMPLE 1 Using Exponents Solution: Write 2 · 2 · 2 in exponential form and evaluate. Slide 4.1-5

5. Solution: - 6 2 Base: Exponent: Base Exponent CLASSROOM EXAMPLE 2 Evaluating Exponential Expressions Evaluate. Name the base and the exponent. Note the difference between these two examples. The absence of parentheses in the first part indicates that the exponent applies only to the base 2, not −2. Slide 4.1-6

6. Objective 2 Use the product rule for exponents. Slide 4.1-7

7. Generalizing from this example suggests the product rule for exponents. Use the product rule for exponents. By the definition of exponents, Product Rule for Exponents For any positive integersmandn,am·an= am + n. (Keep the same base; add the exponents.) Example:62·65= 67 Do not multiply the bases when using the product rule. Keep the same base and add the exponents. For example 62· 65 = 67, not 367. Slide 4.1-8

8. CLASSROOM EXAMPLE 3 Using the Product Rule Solution: Use the product rule for exponents to find each product if possible. The product rule does not apply. The product rule does not apply. Be sure you understand the difference between adding and multiplyingexponential expressions. For example, Slide 4.1-9

9. Objective 3 Use the rule (am)n = amn. Slide 4.1-10

10. Power Rule (a) for Exponents For any positive number integersmandn,(am)n= amn. (Raise a power to a power by multiplying exponents.) Example: Use the rule (am)n = amn. We can simplify an expression such as (83)2 with the product rule for exponents. The exponents in (83)2 are multiplied to give the exponent in 86. Slide 4.1-11

11. CLASSROOM EXAMPLE 4 Using Power Rule (a) Solution: Simplify. Be careful not to confuse the product rule, where 42 · 43 = 42+3 = 45 =1024 with the power rule (a) where (42)3 = 42 · 3 = 46 = 4096. Slide 4.1-12

12. Objective 4 Use the rule (ab)m = am bm. Slide 4.1-13

13. Power Rule (b) for Exponents For any positive integerm,(ab)m=ambm. (Raise a product to a power by raising each factor to the power.) Example: Use the rule (ab)m = ambm. We can rewrite the expression (4x)3 as follows. Slide 4.1-14

14. Power rule (b) does not apply to a sum. For example, but CLASSROOM EXAMPLE 5 Using Power Rule (b) Solution: Simplify. Use power rule (b) only if there is one term inside parentheses. Slide 4.1-15

15. Use the rule . Objective 5 Slide 4.1-16

16. 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: 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. Slide 4.1-17

17. CLASSROOM EXAMPLE 6 Using Power Rule (c) Solution: Simplify. In general, 1n = 1, for any integer n. Slide 4.1-18

18. Rules of Exponents The rules for exponents discussed in this section are summarized in the box. These rules are basic to the study of algebra and should be memorized. Slide 4.1-19

19. Objective 6 Use combinations of rules. Slide 4.1-20

20. CLASSROOM EXAMPLE 7 Using Combinations of Rules Solution: Simplify. Slide 4.1-21

21. Objective 7 Use the rules for exponents in a geometry application. Slide 4.1-22

22. CLASSROOM EXAMPLE 8 Using Area Formulas Write an expression that represents the area of the figure. Assume x>0. Solution: Slide 4.1-23