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  4. If ω (≠ 1) be a cube root of unity and (1 + ω2)n = (1 + ω4)n, then the least positive value of n is

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Q. If $\omega (\ne 1)$ be a cube root of unity and $(1 + \omega^2)^n = (1 + \omega^4)^n,$ then the least positive value of $n$ is

IIT JEEIIT JEE 2004Complex Numbers and Quadratic Equations

Solution:

Given, $(1+\omega^2)^n=(1+\omega^4)^n$
$\Rightarrow (-\omega)^n =(-\omega^2)^n \, [\because w^3=1 $ and $1+\omega+\omega^2=0]$
$\Rightarrow \omega^n =1$
$\Rightarrow n = 3$ is the least positive value of $n$.