If $\alpha$ and $\beta$ are the complex cube roots of unity, then $\alpha^3+\beta^3+\alpha^{-2} imes \beta^{-2}$ is equal to

Question

If $\alpha$ and $\beta$ are the complex cube roots of unity, then $\alpha^3+\beta^3+\alpha^{-2} 	imes \beta^{-2}$ is equal to

Aicharya · Accepted Answer

$\begin{aligned} & \alpha^3+\beta^3+\alpha^{-2} \cdot \beta^{-2}=\alpha^3+\beta^3+\frac{1}{\alpha^2 \beta^2} \ & =\omega^3+\omega^6+\frac{1}{\omega^2 \omega^4}=1+1+1=3\end{aligned}$

If $\alpha$ and $\beta$ are the complex cube roots of unity, then $\alpha^3+\beta^3+\alpha^{-2} \times…

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