    Question

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

A

$128\omega$

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B

$-128\omega$

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C

$128{\omega }^{2}$

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D

$-128{\omega }^{2}$

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Solution

## The correct option is D $-128{\omega }^{2}$Explanation for the correct optionAs we know that when $\omega$ is an imaginary cube root of unity then${\omega }^{3}=1\text{and}1+\omega +{\omega }^{2}=0$ which can be rewritten as $1+\omega =-{\omega }^{2}$Now, the given expression is $\left(1+\omega -{\omega }^{2}{\right)}^{7}$$\begin{array}{rcl}{\left(1+\omega -{\omega }^{2}\right)}^{7}& =& {\left(-{\omega }^{2}-{\omega }^{2}\right)}^{7}\\ & =& {\left(-2{\omega }^{2}\right)}^{7}\\ & =& -128\left({\omega }^{14}\right)\\ & =& -128\left({\left({\omega }^{3}\right)}^{4}·{\omega }^{2}\right)\\ & =& -128\left({\left(1\right)}^{4}·{\omega }^{2}\right)\\ & =& -128{\omega }^{2}\end{array}$Hence, the correct option is (D).  Suggest Corrections  1      Similar questions  Explore more