If $\mathrm{\alpha }, \mathrm{\beta }$ are the imaginary cube roots of unity then ${\mathrm{\alpha }}^4+{\mathrm{\beta }}^4-{(\mathrm{\alpha }+\mathrm{\beta })}^{-1}=$

$\mathrm{\alpha }, \mathrm{\beta }$ are the imaginary cube roots of unity.

$\begin{array}{rcl} & \Rightarrow & \mathrm{\alpha }=\mathrm{w}, \mathrm{\beta } ={\mathrm{w}}^2\end{array}$

$\begin{array}{rcl} & & \\ {\mathrm{\alpha }}^4+{\mathrm{\beta }}^4 -{(\mathrm{\alpha }+\mathrm{\beta })}^{-1} & =& {\mathrm{w}}^4+{\left({\mathrm{w}}^2\right)}^4-\frac{1}{{\mathrm{w}}^2+\mathrm{w}}\end{array}$

$\begin{array}{rcl} & =& w+w^2-\frac{1}{\left(-1\right)}=w+w^2+1=0\end{array}$