What is 0^0?

Date: 11/01/2001 at 17:30:50
From: Molly Shoemaker
Subject: What is 0^0

I know you've answered this before, but I really don't understand your 
answer. What is 0^0?  

We are doing exponents in school and we were talking about how 9^0=1, 
10^0=1, etc., and I asked what 0^0 is. My teacher didn't know so I 
decided to find out.  Your answer to this question in your archives 
confuses me, so could you explain it better?

Date: 11/02/2001 at 01:08:08
From: Doctor Jeremiah
Subject: Re: What is 0^0

Hi Molly,

This following section of the Dr. Math FAQ is where all the
good information is:

  http://www.mathforum.org/dr.math/faq/faq.0.to.0.power.html   

This is why 0^0 is called an inderterminate form:

Anything times zero is zero:

 0^1 = 0       = 0
 0^2 = 0*0     = 0
 0^3 = 0*0*0   = 0
 0^4 = 0*0*0*0 = 0

But as you mentioned earlier, anything to a power of zero is one:

 1^0 = 1
 2^0 = 1
 3^0 = 1
 4^0 = 1

So would 0^0 be 1 or would it be 0? Well, there is no right or wrong 
answer to this, and since there are two right answers, we say that 
it's not answerable. (We call it "indeterminate" because indeterminate 
means that the answer can't be determined.)

There are other things you can do with 0 that are indeterminate; 0/0 
is one of them.

All you can really say about 0^0 is that there is no way to know what 
the answer is.

That's why its indeterminate, but you can sometimes find out what 
answer it might be by taking the limit of the top and dividing it by 
the limit of the bottom. This is called L'Hopital's Rule and is
something you learn much later on.

So sometimes there is an answer, and sometimes you can figure out what 
it might be, but usually there is just no way to know (which is why 
it's called "indeterminate").

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/