It's provable pretty easily from Euler's formula:

e^(pi*i)=-1

(e^(pi*i))^x = (-1)^x

(-1)^x = e^(pi*i*x)

I think it's kind of a nice notation for the unit circle on the complex plane because you don't have to write all the e's,pis, and i's everywhere. And it's easier to work in then radians, for example (-1)^(22.4) is obviously the same as (-1)^(.4) when you mod it by 2 which is .4 of the way to the left of the circle after going around 11 times.. But e^(22.4*i) you would have to think modulo 2*pi which might not be that obvious. It ends up that is 3 times around the circle and .5 of the way to the left of the circle....

I
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