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
It’s just defined that way.
The *multiply X by itself* n *times* “definition” is a simplification of the definition of exponentiation to *natural* powers, not zero.
It is convenient to define it that way because if we multiply two numbers X^n and X^m for natural numbers *n* and *m*, then it is clear that it is equal to X^(n+m). With this, we can relate exponentiation to multiplication as we do multiplication to addition.
In choosing how to define exponentiation to negative integers, we use the property above. Because the natural numbers are closed under addition, for any natural number *k ≠ 1*, there exists *m* such that k = m + 1. Then, because we have defined X^k, we know that X^(k-1) = X^m. Because this is equal to a product of X’s, we know that X^m = X^k / X.
It follows from this that the inverse of a real number X under multiplication is X^(-1), i.e., X^(-1) is the number such that X * X^(-1) = 1 (they “cancel out”).
Applying the relationship above, X^n * X^m = X^(n+m), we have X * X^(-1) = X^1 * X^(-1) = X^(1-1) = X^0 = 1.
I am going to bed…
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