Powers of i

Multiply i by itself again and again. i^1 = i and i^2 = -1. Then i^3 = i^2 × i = -i, and i^4 = i^2 × i^2, which is -1 times -1, so 1. One more factor starts over: i^5 = i^4 × i = i. The powers cycle through four values forever: i, -1, -i, 1.

So for a big power, divide the exponent by 4 and keep the remainder. For i^23: 23 = 4 × 5 + 3, so i^23 = (i^4)^5 × i^3, which is 1 times -i, or -i. A remainder of 1 gives i, 2 gives -1, 3 gives -i, and 0 gives 1.

Why 4? Because i^4 = 1, and multiplying by 1 changes nothing, so every group of four i's drops out. That is why i^40 = 1: 40 is 4 times 10, and no i is left over.