4.1 The Product Rule and Power Rules for Exponents

Objective 1 Use exponents. Slide 4.1-3

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 4.1-4

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

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 indicates that the exponent applies only to the base 2, not −2. Slide Evaluating Exponential Expressions 6 2  CLASSROOM EXAMPLE 2

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

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 Slide 4.1-8

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 Using the Product Rule CLASSROOM EXAMPLE 3

Objective 3 Use the rule (a m ) n = a mn. Slide

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 Use the rule (a m ) n = a mn.

Solution: Simplify. Be careful not to confuse the product rule, where 4 2 · 4 3 = = 4 5 =1024 with the power rule (a) where (4 2 ) 3 = 4 2 · 3 = 4 6 = Slide Using Power Rule (a) CLASSROOM EXAMPLE 4

Objective 4 Use the rule (ab) m = a m b m. Slide

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

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 Using Power Rule (b) CLASSROOM EXAMPLE 5

Objective 5 Use the rule. Slide

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

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

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

Objective 6 Use combinations of rules. Slide

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

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

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