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

Answer: (D)

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

Given:-

$\omega$ is an imaginary cube root of unity.

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

Solution:-

We know that for cube root of unity.

$1+\omega +{\omega }^2=0$ \therefore 1+\omega =-{ \omega }^{ 2 }$

Put the value of $1+\omega$ in the given equation

${(1+\omega -{\omega }^2)}^7={(-{\omega }^2-{\omega }^2)}^7={(-2{\omega }^2)}^7=-128{\omega }^{14}=-128\left(\omega {)}^{14}\right(\omega {)}^2=-128\left({\omega }^3{)}^2\right(\omega {)}^2\{{\omega }^3=1\}=-128(\omega {)}^2$