Answer by Edwin Chen:

Here's a geometric (but non-rigorous) way to think about it:

x^n is the volume of an n-cube with side length x, aka the** number of unit n-cubes you need to make an n-cube with side length x**. For example, eight unit cubes make a cube of side length two (2^3 = 8).

In zero dimensions, **everything is a point**. In particular, the unit 0-cube and the 0-cube of side length x are both just points. Since it only takes one point to make another point, we have x^0 = 1.

Why does a number raised to the zeroth power equal one?

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