The one I like is that for cardinal (counting) numbers, x^y counts the number of functions from a set of y elements to one with x elements. This is very foundational to how powers are defined in the first place, and in many ways even more foundational than addition and multiplication.
If y is empty (regardless of x), then there is exactly one such: the empty function. So x^0 = 1 for all cardinal numbers x including x=0 and all the natural numbers, but also all the infinite cardinalities as well.
From there you can extend to integers, rational numbers, reals, and so on.
The one I like is that for cardinal (counting) numbers, x^y counts the number of functions from a set of y elements to one with x elements. This is very foundational to how powers are defined in the first place, and in many ways even more foundational than addition and multiplication.
If y is empty (regardless of x), then there is exactly one such: the empty function. So x^0 = 1 for all cardinal numbers x including x=0 and all the natural numbers, but also all the infinite cardinalities as well.
From there you can extend to integers, rational numbers, reals, and so on.
Of course, from a pedagogical point of view it may be hard to explain why the “empty function” is actually a function.