Zero to a Negative Exponent
Date: 05/06/2001 at 22:15:44 From: Dave Bailis Subject: Zero to a negative exponent The students and I had the following problem: What is 0^-3 (zero to the negative third power)? We decided that zero to the (any number) exponent should be, by definition, zero; however, the textbook answer key stated that the answer was undefined. We could not decide which operation should be completed first, finding the reciprocal before taking zero to the power, or vice-versa. What do you think about this issue? Is there some order of operations rule that we are missing? Thank you, Dave Bailis
Date: 05/07/2001 at 08:50:04 From: Doctor Peterson Subject: Re: Zero to a negative exponent Hi, Dave (and class). This is not really an order-of-operations issue, because the order itself is unambiguous: you negate the 3, then use it as an exponent. Rather, it is a definition issue. Your question as stated is whether we should calculate 0^-3 as 1 1 ----- or (---)^3 (0^3) 0 But these produce the same answer. The first becomes 1/0, which is undefined. The second requires you to cube an undefined expression, which still is undefined. I think what you meant to say is that you are assuming that zero to ANY power is zero, and you want that to take precedence over any considerations of the meaning of a negative exponent. But your assumption is only valid for positive exponents. My reasoning above derives from this fact the rule for zero to a negative power, namely that it is undefined. Be careful about making definitions. If you want to DEFINE 0 to any power as 0, you have to check your definition for consistency. When we define, for example, any number to the 0 power as 1, we do so not because we think that sounds about right, but because it makes the rules for exponents consistent, as explained in the Ask Dr. Math FAQ: n^0 (any number to zero power) http://mathforum.org/dr.math/faq/faq.number.to.0power.html But before we make even that definition final, we have to check carefully, and restate it to exclude ZERO to the zero power, which is indeterminate (but see our separate FAQ on 0^0): 0^0 (zero to zero power) http://mathforum.org/dr.math/faq/faq.0.to.0.power.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 05/07/2001 at 09:01:45 From: Doctor Jerry Subject: Re: Zero to a negative exponent Hi Dave, One thing you can do to get some guidance in questions like this is to try it on your scientific calculator. Both of mine returned undefined when I gave them either 0^-3 or 0^(-3). In defining exponential functions, one takes care of negative exponents by reducing the problem to the positive exponent case, that is, a^b = 1/a^(-b) where b < 0. So, if b = -3, we consider a^3, and, specifically, 0^3 = 0. So, we would run into an undefined operation. Hence, 0^(-3) is undefined. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/
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