Question

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

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

Answer 1

Answer: (D)

Key Idea : If $\omega$ is a cube root of unity, then
$1+\omega +{\omega }^2=0$ and ${\omega }^3=1$
${\left(l+\omega -{\omega }^2\right)}^7={\left(-{\omega }^2-{\omega }^2\right)}^7$
$\left(\because 1+\omega +{\omega }^2=0\right)$
$={\left(-2{\omega }^2\right)}^7$
$=-2^7\cdot {\omega }^{14}$
$=-128{\left({\omega }^3\right)}^4{\omega }^2$
$=-128{\omega }^2(\because {\omega }^3=1$