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If $\alpha$ and $\beta$ are imaginary cube roots of unity, then $\alpha^4+\beta^4+\frac{1}{\alpha \beta}=$
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Complex cube root of unity are $1, \omega, \omega^2$
Let $\alpha=\omega, \beta=\omega^2$; Then $\alpha^4+\beta^4+\alpha^{-1} \beta^{-1}$ $=\omega^4+\left(\omega^2\right)^4+\left(\omega^{-1}\right)\left(\omega^2\right)^{-1}=\omega+\omega^2+1=0$
Let $\alpha=\omega, \beta=\omega^2$; Then $\alpha^4+\beta^4+\alpha^{-1} \beta^{-1}$ $=\omega^4+\left(\omega^2\right)^4+\left(\omega^{-1}\right)\left(\omega^2\right)^{-1}=\omega+\omega^2+1=0$
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