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Powers of i worksheets

The powers of i do not grow. They go round: i, −1, −i, 1, and back to i. These free printable worksheets ask for powers from i² to i²³, and the answer key works each one by dividing the exponent by 4 and reading off the remainder.

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Powers of i
Powers of i worksheet — Exponents 2 to 23, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The cycle is not a fact to memorise; it is i² = −1 used twice. i³ is i² × i, which is −i, and i⁴ is i² × i², which is (−1)(−1) = 1. Once a power reaches 1, the next multiplication by i starts the pattern again from i, so every fourth power is 1 and nothing new ever appears. A student who can rebuild those four lines does not need to remember them.

Why this sheet works

That turns a large power into a question about division. i¹⁷ is four full laps of the cycle and one step more, so it is i. The key writes it the same way every time — 17 ÷ 4 = 4 remainder 1, so i¹⁷ = i — with the cycle printed above it. The slip it is laid out to catch is a remainder of 0: i¹⁶ lands on 1, the end of a lap, not on i and not on 0.

How this one works

i¹⁷

i¹ = i, i² = −1, i³ = −i, i⁴ = 1
17 ÷ 4 = 4 remainder 1
i¹⁷ = i
  1. The powers of i go round in fours: i, then −1, then −i, then 1, and then i again. Nothing is ever computed here — the question is only which of the four i¹⁷ lands on.
  2. So divide the power by 4 and keep the REMAINDER: 17 ÷ 4 is 4 remainder 1. The 4 whole turns bring it back to where it started and change nothing.
  3. A remainder of 1 means i¹⁷ is the same as i¹, which is i.

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