Question

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

• A. 0
• B. $-1$
• C. 1
• D. $\frac{1}{2}$

Answer 1

The correct option is A 0

The non real complex root of unity are $w,w^2$ such that
$w^3=1$ and
$1+w+w^2=0$
and
$\overline{w}=w^2$
Hence
${\alpha }^4+{\beta }^4+{\alpha }^{-1}.{\beta }^{-1}$
$=w^4+(w^2{)}^4+(\overline{w}\left)\right(\overline{w^2})$
$=w^{3}w+w^8+w^2\overline{w^2}$
$=w+w^6.w^2+|w^2|$
$=w+w^2+1$
$=1+w+w^2$
$=0$