Question

Which of the following is equivalent to the complex number i^15?

Answer 1

`i^(15) = -i`

Explanation:

The basic relation of the complex numbers is:

`i^(2) = -1`

So, we decompose
`i^(15)` in factors of
`i^(2)`.

So:

`i^(15) = i^(2)*i^(2)*i^(2)*i^(2)*i^(2)*i^(2)*i^(2)*i`

Each
`i^(2)` is replaced by -1.

So:

`i^(15) = (-1)*(-1)*(-1)*(-1)*(-1)*(-1)*(-1)*i`

`i^(15) = (-1)^(7)*i`

Any negative value powered to an odd value will be negative. So:

`(-1)^(7) = -1`

`i^(15) = -i`