Question

How do I find the three cube roots of unity?

Answer 1

Although you could use de Moivre's Threorem, or another version of the geometry of the complex plane, it's not needed if you know and apply some algebra:

The cube roots on unity (1) are the complex solutions to
`x^3=1`

`x^3-1=0`

Now factor

Method 1: use `a^3-b^3= (a-b)(a^2+ab+b^2)` to get
`x^3-1=(x-1)(x^2+x+1)`

Method 2: Use the (obvious?) fact that `1` is a zero of `x^3-1` to see that `(x-1)` must be a factor.
Then do the division (long or synthetic) (or use trial and error) to get `x^3-1=(x-1)(x^2+x+1)`

So we need to solve:
`x^3-1=(x-1)(x^2+x+1)=0`.

Use your favorite applicable method to solve: `(x^2+x+1)=0`