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)

$-128{\omega }^2$

$1+w+w^2=0$
$1+w=-w^2$
Hence
$(1+w-w^2{)}^7$
$=(-2w^2{)}^7$
$=-128.w^{14}$
$=-128.w^{12}.w^2$
$=-128.w^2$
Since $w^{3n}=1$ where n is a natural number.