Powers of i

Powers of i

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

Formulas: i, i2, i3, i4

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

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

These formulas are used
to find the higher powers of i.

Example 1: Simplify i23

Simplify the given expressions. i^23

To use the formula [i4 = 1],
divide the exponent 23 by 4.

The quotient is 5.
The remainder is 3.

So 23 = 4⋅5 + 3.

So i23 = i4⋅5 + 3.

i4⋅5 + 3 = i4⋅5i3

Product of powers

i4⋅5 = (i4)5

Power of a power

i4 = 1
i3 = -i

Then (given) = 15⋅(-i).

15 = 1

So (given) = -i.

Example 2: Simplify i86

Simplify the given expressions. i^86

Divide the exponent 86 by 4.

The quotient is 21.
The remainder is 2.

So 86 = 4⋅21 + 2.

So i86 = i4⋅21 + 2.

i4⋅21 + 2 = i4⋅21i2

i4⋅21 = (i4)21

i4 = 1
i2 = -1

Then (given) = 121⋅(-1).

121 = 1

So (given) = -1.