# Powers of *i*

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

## Formula

*i*^{1} = *i**i*^{2} = -1*i*^{3} = -*i**i*^{4} = 1

To solve the powers of *i*,

remember these formulas.

## Proof

*i*^{2} = (√-1)^{2}

Imaginary numbers (*i*)

Then the square root and the square are cancelled.

So (√-1)^{2} = -1.

*i*^{3} = *i*^{2}⋅*i*

= -1⋅*i*

= -*i*

*i*^{3} = *i*^{3}⋅*i*

= -*i*⋅*i*

= -(-1)

= +1

## Examples

Divide the power by 4.

Then (given) = *i*^{(remainder)}.*i*^{5} = *i*^{4⋅1 + 1}

= 1^{1}⋅*i*^{1}

= *i*

Divide the power by 4.

Then (given) = *i*^{(remainder)}.*i*^{82} = *i*^{4⋅20 + 2}

= 1^{20}⋅*i*^{2}

= *i*^{2}

= -1