Why is n^0 always 1 and not 0?

Anything raised to the zeroth power is a difficult thing to get your head around. The easiest explanation (not a full proof) is to look at what happens as we go down in powers of n: n^3=nnn        n^2=(n^3)/n=nn       n^1=(n^2)/n=n From that it follows that n^0=(n^1)/n=n/n=1 So n^0=1. I think the easiest way to think about this conceptually is that, although x+0=x, x0=0 while x*1=1. Funny things happen with 0, which is why you should never consider the expression 0^0 as either equal to 0 or 1! (Or not at this level anyway.)