Question

Let $z = a + ib , b \neq 0$ be complex numbers satisfying $z ^2=\overline{ z } \cdot 2^{1-| z |}$. Then the least value of $n \in N$, such that $z ^{ n }=( z +1)^{ n }$, is equal to

Answer 1

Sol. $\left| z ^2\right|=|\overline{ z }| \cdot 2^{1-| z |} \Rightarrow| z |=1$
$z ^2=\overline{ z } \Rightarrow z ^3=1 $
$\therefore z =\omega \text { or } \omega^2$
$\omega^{ n }=(1+\omega)^{ n }=\left(-\omega^2\right)^{ n }$
Least natural value of $n$ is 6 .