Powers of i

Powers of i

How to solve the powers of i problems: formula, proof, examples, and their solutions.

Formula

i = i, i^2 = -1, i^3 = -i, i^4 = 1

i1 = i
i2 = -1
i3 = -i
i4 = 1

To solve the powers of i,
remember these formulas.

Proof

The proof of i^2 = -1, i^3 = -i, and i^4 = 1

i2 = (√-1)2

Imaginary numbers (i)

Then the square root and the square are cancelled.

So (√-1)2 = -1.

i3 = i2i
= -1⋅i
= -i

i3 = i3i
= -ii
= -(-1)
= +1

Examples

Simplify the given expressions. 1. i^5 2. i^84

Divide the power by 4.
Then (given) = i(remainder).

i5 = i4⋅1 + 1
= 11i1
= i

Divide the power by 4.
Then (given) = i(remainder).

i82 = i4⋅20 + 2
= 120i2
= i2
= -1