1. Chapter 5
Section 1

2. The Product Rule and Power Rules for Exponents
5.1
The Product Rule and Power Rules for Exponents
Use exponents.
Use the product rule for exponents.
Use the rule (am)n = amn.
Use the rule (ab)m = ambn.
Use the rule
Use combinations of rules.
Use the rules for exponents in a geometric application. 2 3 4 5 6 7

3. Objective 1
Use exponents.
Slide 5.1-3

4. Use exponents.
Recall from Section 1.2 that in the expression 52, the number 5 is the base and 2 is the exponent or power. The expression 52 is 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 5.1-4

5. Write 2 · 2 · 2 in exponential form and evaluate.
EXAMPLE 1
Using Exponents
Write 2 · 2 · 2 in exponential form and evaluate.
Solution:
Slide 5.1-5

6. - 2 6 EXAMPLE 2 Evaluating Exponential Expressions
Evaluate. Name the base and the exponent.
Solution: 6 2 -
Base: Exponent:
Base Exponent
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

7. Use the product rule for exponents.
Objective 2
Use the product rule for exponents.
Slide 5.1-7

8. Use the product rule for exponents.
By the definition of exponents,
Generalizing from this example
suggests the product rule for 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: 62 · 65 = 67
Do not multiply the bases when using the product rule. Keep the same base and add the exponents. For example · 65 = 67, not 367.
Slide 5.1-8

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

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

11. Power Rule (a) for Exponents
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.
Power Rule (a) for Exponents
For any positive number integers m and n, (am)n = amn.
(Raise a power to a power by multiplying exponents.)
Example:
Slide

12. EXAMPLE 4 Using Power Rule (a) Simplify. Solution:
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

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

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

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

16. Objective 5
Use the rule
Slide

17. Power Rule (c) for Exponents
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

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

19. 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

20. Use combinations of rules.
Objective 6
Use combinations of rules.
Slide

21. Using Combinations of Rules
EXAMPLE 7
Using Combinations of Rules
Simplify.
Solution:
Slide

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

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