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/