If and are imaginary cube roots of unity, then ^4 + ^4 + 1 = - Vidyadip

MCQ

If $\alpha $ and $\beta $ are imaginary cube roots of unity, then ${\alpha ^4} + {\beta ^4}$ + $\frac{1}{{\alpha \beta }} = $

A
$3$
✓
$0$
C
$1$
D
$2$

✓

Answer

Correct option: B.

$0$

b
(b) 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} + {({\omega ^2})^4} + ({\omega ^{ - 1}})\,\,{({\omega ^2})^{ - 1}} = \omega + {\omega ^2} + 1 = 0$.

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