Question

If $\omega$ is an imaginary fifth root of unity, then find the value of $lo{g}_2|1+\omega +{\omega }^2+{\omega }^3-1/\omega |$.

Answer 1

Here ${\omega }^5=1$. Therefore, ${\omega }^{-1}={\omega }^4$
Also, $1+\omega +{\omega }^2+{\omega }^3+{\omega }^4=0$
$\therefore$ $y=lo{g}_2|1+\omega +{\omega }^2+{\omega }^3-\frac{1}{\omega }|$
$\Rightarrow y=lo{g}_2|1+\omega +{\omega }^2+{\omega }^3-{\omega }^4|$ ($\therefore \left|\omega \right|=1$)
$\Rightarrow y=lo{g}_2|-2{\omega }^4|=lo{g}_{2}2=1$
Ans: 1