Question

If $\omega(\neq 1)$ is a complex cube root of unity and $\left(1+\omega^4\right)^n=\left(1+\omega^8\right)^n$, then the least positive integral value of $n$ is

• A. 2
• B. 3
• C. 6
• D. 12

Answer 1

Answer: (B)

As $ \omega^4=\omega, \omega^8=\omega^2$, we get
$(1+\omega)^n=\left(1+\omega^2\right)^n $
$\Rightarrow \left(-\omega^2\right)^n=(-\omega)^n$
$\Rightarrow \omega^n=1 $
$\therefore n=3$