Let $1, \omega$ and $\omega^2$ be the cube roots of unity. What is the value of $\left(1-\omega+\omega^{-1}\right)^5-2\left(1+\omega-\omega^{-1}\right)^4=$ ?

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Aicharya · Accepted Answer

We have, $ \begin{aligned} & \left(1-\omega+\omega^{-1}ight)^5-2\left(1+\omega-\omega^{-1}ight)^4 \ = & \left(1-\omega+\omega^2ight)^5-2\left(1+\omega-\omega^2ight)^4 \quad\left[\because \omega^{-1}=\omega^2ight] \ = & (-2 \omega)^5-2\left(-2 \omega^2ight)^4 \quad\left[\because 1+\omega+\omega^2=0ight] \end{aligned} $ $\begin{aligned} & =-32 \omega^5-32 \omega^8=-32\left(\omega^5+\omega^8ight) \ & =-32\left(\omega^2+\omega^2ight)=-64 \omega^2=-64 \omega^{-1}\end{aligned}$

Let $1, \omega$ and $\omega^2$ be the cube roots of unity. What is the value of…

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