?In this section we defined a0 = 1. It is important to realize that a definition is

Chapter 7, Problem 91

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In this section we defined a0 = 1. It is important to realize that a definition is neither right nor wrong—it just is. The proper question to ask about a definition is “Is it useful?” What happens if someone decides on an “alternative definition” such as a0 = 7 because 7 happens to be his or her favorite number? What happens when we consider the expression a0 • a4? If you use exponent rule 1 you get a0 • a4 = a0 + 4 = a4. However, if you use this alternative definition, you get a0 • a4 ≟ 7a4. The answer we get from the alternative definition is not consistent with the answer we get from the exponent rule. The exponent rules and this alternative definition cannot coexist. Since we do not want to throw away all the exponent rules, we must modify the alternative definition so that it is consistent with all the exponent rules. Verify that our definition of a0 = 1 is consistent with all five exponent rules.