1. Chapter 5 Section 2

2. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use 0 as an exponent. Use negative numbers as exponents. Use the quotient rule for exponents. Use combinations of rules. Integer Exponents and the Quotient Rule 5.2 2 3 4

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Integer Exponents and the Quotient Rule In all earlier work, exponents were positive integers. Now, to develop a meaning for exponents that are not positive integers, consider the following list. Each time the exponent is reduced by 1, the value is divided by 2 (the bases). Using this pattern, the list can be continued to smaller and smaller integers. From the preceding list, it appears that we should define 2 0 as 1 and negative exponents as reciprocals. Slide 5.2-3

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 1 Use 0 as an exponent. Slide 5.2-4

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. so that the product rule is satisfied. Check that the power rules are also valid for a 0 exponent. Thus we define a 0 exponent as follows. Use 0 as an exponent. The definitions of 0 and negative exponents must satisfy the rules for exponents from Section 5.1. For example, if 6 0 = 1, then and Zero Exponent For any nonzero real number a, a 0 = 1. Example: 17 0 = 1 Slide 5.2-5

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Evaluate. Slide 5.2-6 EXAMPLE 1 Using Zero Exponents

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Use negative numbers as exponents. Slide 5.2-7

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Since and we can deduce that 2 −n should equal Is the product rule valid in such a case? For example, Use negative numbers as exponents. The expression 6 −2 behaves as if it were the reciprocal of 6 2 since their product is 1. The reciprocal of 6 2 is also leading us to define 6 −2 as Negative Exponents For any nonzero real number a and any integer n, Example: Slide 5.2-8

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution : Simplify. Slide 5.2-9 EXAMPLE 2 Using Negative Exponents

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Consider the following: Therefore, Changing from Negative to Positive Exponents For any nonzero numbers a and b and any integers m and n, and Example:and Slide 5.2-10 Use negative numbers as exponents. (cont’d)

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution : Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. We cannot use this rule to change negative exponents to positive exponents if the exponents occur in a sum or difference of terms. For example, would be written with positive exponents as Slide 5.2-11 EXAMPLE 3 Changing from Negative to Positive Exponents

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 3 Use the quotient rule for exponents. Slide 5.2-12

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. We know that Use the quotient rule for exponents. Notice that the difference between the exponents, 5 − 3 = 2, this is the exponent in the quotient. This example suggests the quotient rule for exponents. Quotient Rule for Exponents For any nonzero real number a and any integer m and n, (Keep the same base; subtract the exponents.) Example: Slide 5.2-13

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. Slide 5.2-14 EXAMPLE 4 Using the Quotient Rule

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The product, quotient, and power rules are the same for positive and negative exponents. Slide 5.2-15 Use the quotient rule for exponents.

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 4 Use combinations of rules. Slide 5.2-16

17. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Simplify. Assume that all variables represent nonzero real numbers. Slide 5.2-17 EXAMPLE 5 Using Combinations of Rules