Question

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

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

Answer 1

The correct option is D

$-128{\omega }^2$

Explanation for the correct option

As 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 $(1+\omega -{\omega }^2{)}^7$

$\begin{array}{rcl}\therefore {\left(1+\omega -{\omega }^2\right)}^7& =& {\left(-{\omega }^2-{\omega }^2\right)}^7\\ & =& {\left(-2{\omega }^2\right)}^7\\ & =& -128\left({\omega }^{14}\right)\\ & =& -128\left({\left({\omega }^3\right)}^4·{\omega }^2\right)\\ & =& -128\left({\left(1\right)}^4·{\omega }^2\right)\\ & =& -128{\omega }^2\end{array}$

Hence, the correct option is (D).