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

for any real number x, a definition of x^(n), where n is an integer superior or equal to 1, is : x.x.x … where x appears n times

a more general definition of x^(n), where n can be any real number, is: e ^(n ln(x))
(apologies about the parenthesis falling appart, editor issue)

where e is the Euler’s number (approx 2.71828) and ln is the natural logarithm.

This definition encompasses the first one. Meaning that it gives the same result that the first one, in the case of n being an integer superior or equal to 1.

Now, for any x, and for n = 0, x^(0) = e^(0 ln(x)) = e^(0) = 1