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

The correct option is D $-128{\omega }^2$

Since $w$ is the imaginary cube root of unity
$w^3=1$ and
$1+w+w^2=0$ ...(i)
Hence
$(1+w-w^2{)}^7$
$=\left(\right(1+w)-w^2{)}^7$
$=(-w^2-w^2{)}^7$ ...from i
$=(-2w^2{)}^7$
$=-128w^{14}$
$=-128w^{12}.w^2$
$=-128(w^3{)}^4.w^2$
$=-128w^2$ ...since $w^3=1$.