In polar form, 8 = 8(cos 0° + j sin 0°).

There are 3 roots, so they will be `θ = 120°` apart.

Using DeMoivre's Theorem:

an = rn(cos + j sin ),

First root:

`8^(1"/"3)=8^(1"/"3)(cos\ 0^text(o)/3+j\ sin\ 0^text(o)/3)`

`=2(cos\ 0^text(o)+j\ sin0^text(o))`

`=2`

Second root:

Add `120°` to the first root:

81/3(cos 120° + j sin 120°) = −1 + 1.732j

Third root:

Add `120°` to the second root:

81/3(cos 240° + j sin 240°) = −1 − 1.732j

3 complex roots of a cubic equation


So the 3 cube roots of `8` are:

`2`, `-1 + 1.732j`, and `-1 - 1.732j`.

To see if the roots are correct, raise each one to power `3` and multiply them out.

Get the Daily Math Tweet!
IntMath on Twitter