eli5: why is x⁰ = 1 instead of non-existent?

Question

It kinda doesn’t make sense.
x¹= x

x² = x*x

x³= x*x*x

etc…

and even with negative numbers you’re still multiplying the number by itself

like (x)-² = 1/x² = 1/(x*x)

In: 1797

Answer 1

I think that best way to see it, is to just halve the power and see where it gets you.

2^4 = 16

2^2 = 4

2^1 = 2

2^(1/2) = 1.414

2^(1/4) = 1.189

2^(1/8) = 1.091
.
.
.
2^(1/1000) = 1.001

The value is approaching 1

If you then flip it over to negative exponent (-1/1000, -1/8, -1/4 …) you will see it continues past 1 into smaller values. Making 0 exponent undefined would leave an undefined value in otherwise continuous function.