Question

Solve the equation $z^3=\overline{z}(z\ne 0).$

• A. $\pm 1$
• B. $\pm i$
• C. $\pm (1+i)$
• D. $\pm (1-i)$

Answer 1

Answer: (A), (B)

The correct options are
A $\pm 1$
B $\pm i$

As $z\ne 0,\left|z^3\right|=\left|\overline{z}\right|$
$⟹\left|z\right|=1$
$\therefore z^3=\overline{z}$
$⟹z^4=z\overline{z}={\left|z\right|}^2=1$
$⟹z=\pm 1,\pm i$
Ans: A,B