Why x to the power 0 equals to 1 proove mathematically?

Answer 1

It is not possible to "prove mathematically" such a statement; x0 = 1 is really a definition, but one that makes sense.

One reason it is defined this way is so that certain laws of exponents continue making sense for different numbers. For example, x2 times x3 = (x times x) (x times x times x) = x5; in other words, you can just add the exponents: 2 + 3 = 5.

In the case of zero, you would expect, for example, that x5 times x0 = x5+0 = x5, so it immediately follows that x0 must be 1. But this is only an explanation why this definition is reasonable; not a proof. You really can't prove that x0 = 1.

By the way, 00 isn't defined.