Definition of Zeroth Power

Question

What is the definition of raising a number to the zeroth power ($x^0$)? I know that many people say that "anything raised to the zeroth power is one" but this is clearly not true since $0^0$ is $undefined$. How then do mathematicians define $x^0$ such that for all real numbers not equal to $0$, $x^0=1$?

Answer 1

Depends how strong of a proof you are looking for, but one way you could think of it is $$x^0 = x^{1-1} = \frac{x^1}{x^1} = \frac{x}{x} = 1$$ It may not intuitively be equal to $1$, but it is necessarily equal to $1$.

Answer 2

$$ x^n x^0 = x^{n+0} = x^n$$ Hence $x^0 = 1$.