Question

Find the number of complex numbers which are conjugate of their own cube.

Answer 1

given that $z^3=\overline{z}$ ...(1)
taking modules, we get $\left|z^3\right|=\left|z\right|$
$\Rightarrow \left|z\right|=0$ ...(2)
and ${\left|z\right|}^2=1$
$\Rightarrow \overline{z}=\frac{1}{z}$ ...(3)
Substituting (3) in (1), we get $z^4=1$ ...(4)
from (2) & (4), we can say that number of complex number satisfying (1) are $5$.

Ans: 5