Definition of Zeroth Power [duplicate]

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 $0$, $x^0=1$?

marked as duplicate by 6005, Community♦Oct 12 '15 at 3:47

• We mathematicians bend the rules and change them anyway we want:) Ok, essentially, we accept $x^0=1$ for all $x$ not zero There are different levels of proof for this. – imranfat Oct 12 '15 at 3:43
• Actually, many mathematicians define $0^0=1$. Ter are strong reasons to do this - any empty product is $1$. The product of $0$ 1's is the same as the product of $0$ 0's. – Thomas Andrews Oct 12 '15 at 3:47
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 $1$.
$$x^n x^0 = x^{n+0} = x^n$$ Hence $x^0 = 1$.