Powers of i

i is defined by i² = -1. Every higher power is built from that one fact. i¹ = i, i² = -1, i³ = i² times i = -i, and i⁴ = i² times i² = (-1)(-1) = 1. Once you hit i⁴ = 1, the pattern resets: i⁵ is just i⁴ times i, which is 1 times i = i again. So the four values i, -1, -i, 1 repeat forever in that order, because multiplying by i⁴ never changes anything.

That makes exponents on i a question of leftovers, not full multiplication. Any exponent n splits into whole groups of 4 plus a remainder: n = 4q + r, where r is 0, 1, 2, or 3. Each group of 4 contributes a factor of i⁴ = 1, so it vanishes, and the whole power of i equals i^r. Finding i⁵⁷ means finding 57 divided by 4, which is 14 remainder 1, so i⁵⁷ = i¹ = i. No repeated multiplication needed.

This idea connects to the rest of complex number arithmetic on the test: once you can reduce any power of i to one of four values, you can simplify expressions, multiply complex numbers, and handle problems mixing real and imaginary parts without getting bogged down. It is a small mechanical skill, but it shows up anywhere an exponent on i appears, so spotting the remainder fast is the whole game.