system
Given the following post, respond as if you're commenting on Reddit:
human
What does x^(Pi) actually represent and how can it be interpreted?

I know that x^n = xxx....x (n times) and something like x^(3/2) is the square root of x^3 but how does one interpret something like x^pi for example. I'm 1st year undergrad and this question just popped into my mind.
gpt
Well, the idea that exponentiation is iterated multiplication (x^n = xxx....x) is just the first step. Indeed, iterated multiplication obviously has to be for integer powers only. There are, however, ways to extend what we mean by power in such a way that we conserve all of the nice properties of exponentiation. So for example, this is why it feels natural to you that x^(1/2) = sqrt(x), despite the fact that you cannot multiply x by itself "half a time". It is because you recognize that what matters is that we conserve the laws by which powers behave, for example here that x^a \* x^b = x^(a+b) and (x^(a))^b = x^(a*b). As it turns out, there is a way to define exponentiation of the form x^y for all real y, provided that x is strictly superior to 0. For that, we use the logarithm and exponential functions. Since e^ln(x) = x and ln(x^(y))=y ln(x), we can see that x^y = e^(y ln x), provided that ln(x) exists (thus x>0). So, to answer you question, x^(pi) = e^(pi ln x).