Verified answer

Yes, it actually has everything to do with DeMoivre's theorem. Write i in polar form: the radius is 1 and the angle is 90 degrees, so i=1*(cos 90 +i sin 90). Now, i^63 =1^63 (cos(90*63)+i sin(90*63)) by DeMoivre's theorem. Use periodicity of sine and cosine to find that i^63 =cos(270)+i sin(270) =-i.

Powers of i have a repeating pattern. For all integers n:

i^1, and i ^(1+4n) = i

i^2, and i^(2+4n) = -1

i^3, and i^(3+4n) = -i

i^4, and i^(4n) = 1

63 is 3 + 4*15 so the answer is -i