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)

$-128{\omega }^2$

$\omega$ is the cube root of unity, implies $1+\omega +{\omega }^2=0$
$\Rightarrow {\omega }^2=-1-\omega$
Thus, the expression can be simplified as

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