<PRE><B>Question 443558</B>
Note that i^4 = 1, so we can factor lots of terms out:


*[tex \LARGE i^{63} = (i^4)^{15}(i^3) = i^3 = -i]


i raised to some power carries a cyclic pattern, which corresponds to the rotation of points around the complex plane (1, i, -1, -i). The following pattern holds, where k is some integer:

*[tex \LARGE i^{4k} = i^0 = 1]
*[tex \LARGE i^{4k + 1} = i^1 = i]
*[tex \LARGE i^{4k + 2} = i^2 = -1]
*[tex \LARGE i^{4k + 3} = i^3 = -i]
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