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human
[mathematics] Shouldn't any constant raised to the power of an irrational number become an irrational? Ex 5^pi. If so, why would e^(pi*i) =-1?

Here's my thinking. Since an irrational number can't be expressed as a/b, how can we determine any constant raised to an irrational number? x^1.5 will be x^3/2, or the sqrt(x^3). But with an irrational number, it can't be represented that way, so how would it be computed? Sure you could cut it off at 50 decimal places but that seems rather arbitrary. There's no way you'll ever be a rational number. 

So into e^pi*i, How can it possibly equal 
-1 on the nose? Even expressed as a infinite sum it doesn't seem it should be rational, much less an integer. It could be -. 9999 repeating but that wouldn't be irrational. Anyway, just a bit confused hear, if anyone could explain it with intro level Calculus I would appreciate it.
gpt
Conjecture: There exists a that an irrational to the power of an irrational is rational.

Proof: sqrt(2) is irrational. sqrt(2)^sqrt(2) is either rational or irrational.If it is rational our work is done, if it is not raise sqrt(2)^sqrt(2) to the power of sqrt(2), then (sqrt(2)^sqrt(2) )^sqrt(2) = sqrt(2)^(sqrt(2)^) ^\* ^sqrt(2)) = sqrt(2)^2 =2

Where the first step follows by laws of exponentiation.

There, we don't even need anything fancy like imaginary numbers. In general if a and b are irrational, a^b need not be irrational, and it can be an integer.