X² is X times X.
X¹ is X.
In X^Y Y is the factor that decides how many Xs are getting multiplied.
So why is X⁰ 1? It comes out of nowhere. Zero Xs are getting multiplied, so there’s no numbers being multiplied, meaning there’s nothing.
In: Mathematics
As you say, exponentiation is defined (at first) by “repeated multiplication.” So:
* X^3 = X * X* X
* X^2 = X * X
* X^1 = X
So that definition makes sense for what we call the *natural numbers* or the *counting numbers.* But when you try to extend this concept to other numbers, things get a bit unsatisfying. If you say that X^0 = 0, that means that the multiplication and division pattern in the previous sequence **stops working**. By that what I mean is:
* To get from X^1 to X^2, you multiply by an extra “X.”
* To get from X^2 to X^3, you multiply by an extra “X.”
If X^0 = 0, then you *can’t get* from X^0 to X^1 through multiplication. Uh oh! That property of exponents is one of the core properties that makes them useful, as it turns out! So, rather than break the pattern, we *define X^0* so that it maintains it:
* To get from X^0 to X^1, you multiply by an extra “X,” so X^0 must be equal to 1.
This also then allows us to define negative exponents (X^(-1) = 1/X^(1) because to get from X^(-1) to X^0 you need to multiply by an extra “X”) and fractional exponents (a bit more complicated to explain, but it follows another pattern of exponents).
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tl;dr – “Repeated Multiplication” is just the starting point for exponents. The myriad of other properties that exponents have end up being more valuable than that starting point, so other non-standard exponential values get defined in order to ensure those properties stay accurate, no matter the exponent!
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