If $\omega$ is a complex cube root of unity, the $\left[\frac{51+73 \omega+87 \omega^2}{73+87 \omega+51 \omega^2}+\frac{51+73 \omega+87 \omega^2}{87+51 \omega+73 \omega^2}\right]=$

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Aicharya · Accepted Answer

$\begin{aligned} & \left[\frac{51+73 \omega+87 \omega^2}{73+87 \omega+51 \omega^2}+\frac{51+73 \omega+87 \omega^2}{87+51 \omega+73 \omega^2}ight]^{15} \ & =\left[\begin{array}{c}\frac{51 \omega^2+73 \omega^3+87 \omega^4}{73+87 \omega+51 \omega^2} 	imes \frac{1}{\omega^2} \ \quad+\frac{51 \omega+73 \omega^2+87 \omega^3}{87+51 \omega+73 \omega^2} 	imes \frac{1}{\omega}\end{array}ight]^{15} \ & =\left[\frac{73+87 \omega+51 \omega^2}{73+87 \omega+51 \omega^2} 	imes \frac{\omega}{\omega^3}+\frac{87+51 \omega+73 \omega^2}{87+51 \omega+73 \omega^2} 	imes \frac{\omega^2}{\omega^3}ight]^{15} \ & =\left(\omega+\omega^2ight)^{15}=(-1)^{15}=-1\end{aligned}$

If $\omega$ is a complex cube root of unity, the $\left[\frac{51+73 \omega+87 \omega^2}{73+87 \omega+51…

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