If $\alpha$ and $\beta$ are imaginary cube roots of units then value of $\alpha^4+\beta^{28}+\frac{1}{\alpha \beta}$ is

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Aicharya · Accepted Answer

Let $\alpha=\omega, \beta=\omega^2$ Now $\alpha^4+\beta^{28}+\frac{1}{\alpha \beta}$ $\begin{aligned} & =\omega^4+\left(\omega^2ight)^{28}+\frac{1}{\omega \omega^2} \ & =\omega^4+\omega^{56}+\frac{1}{\omega^3} \ & =\omega+\omega^2+\frac{1}{1} \ & \left.=0 \quad \quad \quad 	ext { [as } \omega^3=1ight]\end{aligned}$

If $\alpha$ and $\beta$ are imaginary cube roots of units then value of $\alpha^4+\beta^{28}+\frac{1}{\alpha…

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