If $\omega$ is a complex cube root of unity, then the value of $(1+\omega -{\omega }^2{)}^7$ is

Solution

1. Given that $\omega$ is a complex cube root of unity, we know $1+\omega +{\omega }^2=0$, which implies $1+\omega =-{\omega }^2$.
2. Substitute this into the expression: $$\begin{array}{rl}(1+\omega -{\omega }^2{)}^7& =(-{\omega }^2-{\omega }^2{)}^7\\ & =(-2{\omega }^2{)}^7\end{array}$$.
3. Calculate the power: $$\begin{array}{rl}(-2{\omega }^2{)}^7& =(-2{)}^7\cdot ({\omega }^2{)}^7\\ & =-128\cdot {\omega }^{14}\end{array}$$.
4. Since ${\omega }^3=1$, we simplify $$\begin{array}{rl}{\omega }^{14}& =({\omega }^3{)}^4\cdot {\omega }^2\\ & =1^4\cdot {\omega }^2\\ & ={\omega }^2\end{array}$$.
5. Thus, the expression evaluates to $-128{\omega }^2$.
6. Hence the answer is (B).