Powers with Exponents in the Natural Numbers with Zero

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\cdot$ be a binary operation on a set $X$, written multiplicatively, let $x\in X$, and let $n\in\mathbb{N}_{0}$.

¶ If $n\in\mathbb{N}$, the $n$-th power of $x$ is the finite product $x^{n}=\prod_{k=1}^{n}x$.

¶ If $\cdot$ has a neutral element, which is unique by A Binary Operation Has at Most One Neutral Element §unique and written $1$, then $x^{0}=1$.