Question

If $\alpha$ and $\beta$ are the complex cube roots of unity, then ${\alpha }^4+{\beta }^4+{\alpha }^{-1}+{\beta }^{-1}$.

Answer 1

$\alpha$ and $\beta$ are the complex cube roots of unity.

Hence,

$\alpha =\omega ,\beta ={\omega }^2$

Since,

$\Rightarrow {\alpha }^4+{\beta }^4+{\alpha }^{-1}+{\beta }^{-1}$

$\Rightarrow {\alpha }^4+{\beta }^4+\frac{1}{\alpha }+\frac{1}{\beta }$

$\Rightarrow {\alpha }^4+{\beta }^4+\frac{\alpha +\beta }{\alpha \beta }$

Therefore,

$\Rightarrow {\omega }^4+({\omega }^2{)}^4+\frac{\omega +{\omega }^2}{\omega {\omega }^2}$

$\Rightarrow \omega +{\omega }^8+\frac{-1}{{\omega }^3}$

$\Rightarrow \omega +{\omega }^2+\frac{-1}{1}$

$\Rightarrow -1-1$

$\Rightarrow -2$

Hence, this is the answer.