Question

Why is that when you square any number by zero that it equals one?

Answer 1

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

Answer 2

In general, a^n/a^m = a^(n-m) and a/a = 1. We can use these facts to prove that x^0 = 1 so long as x isn't 0.

First, state the obvious: 1 = 1

Next, since any non-zero number divided by itself is one: 1 = a^n/a^n

(It doesn't change how the equation looks, but for the sake of being thorough, you could subsitute (a^n/a^n) in place of 1 in the original equation.)

Then, since dividing like bases requires that you subtract their exponents:
a^n/a^n = a^(n-n) = a^0

Substitute (a^0) in for (a^n/a^n) and you obtain: a^0 = 1

There are two reasons "a" cannot be 0 in this proof: firstly, raising 0 to non-zero powers would still result in zero, so "a" being 0 would cause division by zero in the initial theorems we used, and secondly, 0^0 is considered undefined in itself.