Question

If $\omega$ is an imaginary cube root of unity, then $\left(1+\omega-\omega^{2}\right)^{7}$ equals :

• A. $128 \omega$
• B. $-128 \omega$
• C. $128 \omega^{2}$
• D. $-128 \omega^{2}$

Answer 1

Answer: (D)

Key Idea : If $\omega$ is a cube root of unity, then
$1+\omega+\omega^{2}=0 $ and $ \omega^{3}=1 $
$\left(l+\omega-\omega^{2}\right)^{7}=\left(-\omega^{2}-\omega^{2}\right)^{7}$
$\left(\because 1+\omega+\omega^{2}=0\right)$
$=\left(-2 \omega^{2}\right)^{7} $
$=-2^{7} \cdot \omega^{14} $
$=-128\left(\omega^{3}\right)^{4} \omega^{2} $
$=-128 \omega^{2} \left(\because \omega^{3}=1\right.$