To simplify the nth power of i:
Recall that the values of powers of i repeat in a cycle of length 4, so n (mod 4) corresponds to the place of your power in this cycle.
Compute n (mod 4). In other words, find the remainder of n divided by 4.
Determine in as follows:
If n (mod 4) = 0, then in = 1;
If n (mod 4) = 1, then in = i;
If n (mod 4) = 2, then in = -1; and
If n (mod 4) = 3, then in = -i.
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