powers of 0

Could somebody please explain to me (in laymans terms) why a number raised to the 0 power is 1? If you can type an explanation or link me to a simple article, that'd be great.

Thanks!
-Anon

Dr. Math says:


       3^4
   1 = --- = 3^(4-4) = 3^0
       3^4

Let's consider a few examples.  We could choose any base number to do this with but 4 sounds good.  We'll try several examples with the exponent getting closer to 0 and see what results.

4^1 = 4
4^(.5) = 4^(1/2) = sqrt(4) = 2
4^(.25) = 4^(1/4) = 1.14

As you start taking larger roots of 4, you get smaller and smaller numbers.  As you can see as the exponent get's closer to zero the value gets closer to 1.  You can try this with other bases and more exponents, on either side of 0.

1=(x^n)/(x^n)=x^(n-n)=x^0 ,where x is any real number(i think the more general term would be belongs to any field)

The law a^n * a^m = a^(n+m) holds whenever m and n are any positive integers, for obvious reasons.  (For example, a^2 * a^3 = a*a * a*a*a = a^5 = a^(2+3).)  If you want it to hold when m or n is also 0, you need to agree that a^0 = 1.  If you want a^0 to be something other than 1, then the law breaks when m or n is zero, and that's a pain in the ass.

Similarly, if you want it to work for negative m or n, you need to agree that a^(-1) = 1/a.