Why number raise to zero equals to one?

Answer 1

The simplest way to understand this is to consider what happens to powers of numbers as the exponent is increased or decreased. If you take any number, call it "X" and raise it to a power, let's use 4 as an example, you express it as X4. Consider what would lower that number by 1, giving you X3. For it to become X3, you would simply divide X4 by X. In other words:

X3 = X4 / X

or more generically,

Xn / X = X(n - 1)

So what happens if you logically follow that pattern down to the smallest value of n? Well, consider our example:

X4 / X = X3

X3 / X = X2

X2 / X = X1

X1 / X = X0

X1 is equal to X, no matter what the value of X is. This means that according to our last statement, X0 = X / X. Of course X / X is always equal to one, as long as X is not equal to 0. That means that X0 must equal 1 as well, for any non-zero value.

00 is the one exception here, as that value is undefined. The reason for this is clear if you consider what it means. As stated above, X0 is equal to X / X. In the case of X = 0 though, that would mean it's equal to 0 / 0. That value is undefined, and therefore 00 is as well.