If $\omega$ is the complex cube root of unity, the $\left(3+5 \omega+3 \omega^2\right)^2+\left(3+3 \omega+5 \omega^2\right)^2=$

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Aicharya · Accepted Answer

$\begin{aligned} & \left(3+5 \omega+3 \omega^2ight)^2+\left(3+3 \omega+5 \omega^2ight)^2 \ & =\left(3+3 \omega+3 \omega^2+2 \omegaight)^2+\left(3+3 \omega+3 \omega^2+2 \omega^2ight)^2 \ & =\left[3\left(1+\omega+\omega^2ight)+2 \omegaight]^2+\left[3\left(1+\omega+\omega^2ight)+2 \omega^2ight]^2 \ & =(3(0)+2 \omega)^2+\left[3(0)+2 \omega^2ight]^2=4 \omega^2+4 \omega^4 \ & =4 \omega^2\left(1+\omega^2ight)=4 \omega^2(-\omega) \ & =-4 \omega^2=-4\end{aligned}$

If $\omega$ is the complex cube root of unity, the $\left(3+5 \omega+3 \omega^2\right)^2+\left(3+3 \omega+5…

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