# Thread: Find all complex numbers z

1. ## Find all complex numbers z

Find all complex numbers z so that z3 = -8i.

I know I have to do something with polar form, but I'm really lost.

2. ## Re: Find all complex numbers z

If $z=re^{i\varphi}$, then $z^3=r^3e^{3i\varphi}$. Note that $-i=e^{i(3\pi/2)}$. So you need to find all $0\le\varphi<2\pi$ such that $3\varphi\equiv3\pi/2\pmod{2\pi}$. There are three such $\varphi$. And, of course, you need to find $r$.

3. ## Re: Find all complex numbers z

Originally Posted by biggerleaffan
Find all complex numbers z so that z3 = -8i.
I know I have to do something with polar form, but I'm really lost.
This is posted in basic algebra forum. Reply #2 is correct but advanced.

Notation: $r\exp(i\theta)=r\cos(\theta)+i~r\sin(\theta)~.$ So ${\left[ {r\exp \left( {i\theta } \right)} \right]^n} = {r^n}\exp \left( {i\theta n} \right)$

Let $\rho = 2\exp \left( {\frac{{ - i\pi }}{6}} \right)$, so $\rho^3=-8i$. We have one root.

Let $\xi = \exp \left( {\frac{{2\pi i}}{3}} \right)$, then you need to show that $\rho\cdot\xi~\&~\rho\cdot\xi^2$ are the other two roots.