42 CHAPTER R Review of Basic Concepts Zero as an Exponent A zero exponent is defined as follows. Zero Exponent For any nonzero real number a, a0 =1. That is, any nonzero number with a zero exponent equals 1. To illustrate why a0 is defined to equal 1, consider the product an # a0, for a≠0. We want the definition of a0 to be consistent so that the product rule applies. Now apply this rule. an # a0 = an+0 = an The product of an and a0 must be an, and thus a0 is acting like the identity element 1. So, for consistency, we define a0 to equal 1. (00 is undefined.) EXAMPLE 3 Using the Definition of a0 Evaluate each expression. (a) 40 (b) 1-420 (c) -40 (d) -1-420 (e) 17r20 SOLUTION (a) 40 = 1 Base is 4. (b) 1-420 = 1 Base is -4. (c) -40 = -1402 = -1 Base is 4. (d) -1-420 = -112 = -1 Base is -4. (e) 17r20 = 1, r ≠0 Base is 7r. S Now Try Exercise 35. Negative Exponent Let a be a nonzero real number and n be any integer. a−n = 1 an Negative Exponents and the Quotient Rule Suppose that n is a positive integer, and we wish to define a-n to be consistent with the application of the product rule. Consider the product an # a-n, and apply the rule. an # a-n = an+1-n2 Product rule: Add exponents. = a0 n and -n are additive inverses. = 1 Definition of a0 The expression a-n acts as the reciprocal of an, which is written 1 an. Thus, these two expressions must be equivalent. CAUTION The expressions mn2 and 1mn22 are not equivalent. The second power rule can be used only with the second expression. 1mn22 = m2n2
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