Complex Numbers 5 Question 9

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

(a) $128 w$

(b) $-128 \omega$

(c) $128 \omega^{2}$

(d) $-128 \omega^{2}$

(1998, 2M)

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Answer:

Correct Answer: 10. (d)

Solution:

  1. $\left(1+\omega-\omega^{2}\right)^{7}=\left(-\omega^{2}-\omega^{2}\right)^{7}$

$\left[\because 1+\omega+\omega^{2}=0\right]$

$$ =\left(-2 \omega^{2}\right)^{7}=(-2)^{7} \omega^{14}=-128 \omega^{2} $$



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