Anything to the 0 Power

Date: 11/21/97 at 16:00:46
From: Jenna M Gambino
Subject: N^0

Dear Dr. Math,

Everyone in my 7th grade class wants to know the answer to this 
question: Why is anything to the 0 power 1?

No one in my class knows the answer, not even my teacher. She asked me 
to e-mail Dr. Math and we are eagerly waiting to hear the answer. 

Thanks for your time,

Jenna

Date: 11/24/97 at 14:44:56
From: Doctor Steven
Subject: Re: N^0

Look at the powers of any number, say 8:

   8^1 =    8
   8^2 =   64
   8^3 =  512
   .
   .
   .

To get from 8^3 to 8^2 we have to divide by 8. And to get from 8^2 to
8^1 we have to divide by 8 again.  The logical idea is to continue 
doing this so we get:

  to go from 8^1 to 8^0 we should divide by 8.

So 8^0 = 8^1/8 = 8/8 = 1. This works for any nonzero number. 0^0 is 
undefined. 

We can even continue this process to create negative exponents:

  to go from 8^0 to 8^(-1) we divide by 8

So 8^(-1) = 8^0/8 = 1/8.

Now we have this:

          .
          .
          .
   8^3    = 512
   8^2    =  64
   8^1    =   8
   8^0    =   1
   8^(-1) =   1/8
   8^(-2) =   1/64
   8^(-3) =   1/512
          .
          .
          .

Hope this helps.

For more, see the Dr. Math FAQ:

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

-Doctor Steven,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/