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Chapter 5 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Use 0 as an exponent. Use negative numbers as exponents. Use the quotient rule for exponents. Use combinations of rules. Integer Exponents, and Quotient Rule 1 1 4 4 3 3 2 25.25.2

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 5.2 - 3 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.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Slide 5.2 - 4 Use 0 as an exponent.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 rule for exponents from Section 5.1. For example if 6 0 = 1, then and Slide 5.2 - 5 For any nonzero real number a, a 0 = 1. Example: 17 0 = 1

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using Zero Exponents Solution: Slide 5.2 - 6 Evaluate.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Use negative numbers as exponents. Slide 5.2 - 7

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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, if we multiply Slide 5.2 - 8 The expression 6 −2 behaves as if it were the reciprocal of 6 2 : Their product is 1. The reciprocal of 6 2 is also, leading us to define 6 −2 as. This is a particular case of the definition of negative exponents. For any nonzero real number a and any integer n, Example:

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using Negative Exponents Solution: Slide 5.2 - 9 Simplify.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use negative numbers as exponents. Consider the following: Slide 5.2 - 10 For any nonzero numbers a and b and any integers m and n, and Therefore, Example:and

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Changing from Negative to Positive Exponents Slide 5.2 - 11 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.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Use the quotient rule for exponents. Slide 5.2 - 12

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 5.2 - 13 For any nonzero real number a and any integer m and n, (Keep the same base; subtract the exponents.) Example:

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

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.2 - 15 The product, quotient, and power rules are the same for positive and negative exponents.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Use combinations of rules. Slide 5.2 - 16

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

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