4.2 Integer Exponents and the Quotient Rule

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 20 as 1 and negative exponents as reciprocals. Slide 4.2-3

Objective 1 Use 0 as an exponent. Slide 4.2-4

Use 0 as an exponent. Zero Exponent The definitions of 0 and negative exponents must satisfy the rules for exponents from Section 4.1. For example, if 60 = 1, then and 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. Zero Exponent For any nonzero real number a, a0 = 1. Example: 170 = 1 Slide 4.2-5

CLASSROOM EXAMPLE 1 Using Zero Exponents Evaluate. Solution: Slide 4.2-6

Use negative numbers as exponents. Objective 2 Use negative numbers as exponents. Slide 4.2-7

Use negative numbers as exponents. Since and we can deduce that 2−n should equal Is the product rule valid in such a case? For example, The expression 6−2 behaves as if it were the reciprocal of 62 since their product is 1. The reciprocal of 62 is also leading us to define 6−2 as Negative Exponents For any nonzero real number a and any integer n, Example: Slide 4.2-8

Using Negative Exponents CLASSROOM EXAMPLE 2 Using Negative Exponents Simplify. Solution: Slide 4.2-9

Changing from Negative to Positive Exponents Use negative numbers as exponents. (cont’d) 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 4.2-10

Changing from Negative to Positive Exponents CLASSROOM EXAMPLE 3 Changing from Negative to Positive Exponents Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. Solution: 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 4.2-11

Use the quotient rule for exponents. Objective 3 Use the quotient rule for exponents. Slide 4.2-12

Use the quotient rule for exponents. We know that 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 4.2-13

Using the Quotient Rule CLASSROOM EXAMPLE 4 Using the Quotient Rule Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. Solution: Slide 4.2-14

Use the quotient rule for exponents. The product, quotient, and power rules are the same for positive and negative exponents. Slide 4.2-15

Use combinations of rules. Objective 4 Use combinations of rules. Slide 4.2-16

Using Combinations of Rules CLASSROOM EXAMPLE 5 Using Combinations of Rules Simplify. Assume that all variables represent nonzero real numbers. Solution: Slide 4.2-17