complex numbers - how to compute $|a-ib|^2$ if given $(a-ib)^3$

If i know that, for example, $(a-ib)^3=5+4i$ - how can i compute the value of $|a-ib|^2$ ?

I can take modulus of $5+4i$ which is $\sqrt{5^2+4^2}$ but i don't know what i'm getting here.

I don't know what's a third root of a complex number here.

Any help?

Thanks.

Answer

We know that $|z\cdot w|=|z|\cdot|w|$, and so in general $|z^n|=|z|^n$.

Therefore, since you know $z^3$, you can take its modulus to get $|z^3|=|z|^3$, and raising that to the $2/3$ power gives $|z|^2$.