Question

The value of ${(1-\omega +{\omega }^2)}^5+{(1+\omega -{\omega }^2)}^5$ where $\omega$ and ${\omega }^2$ are the complex cube roots of unity, is

• A. 0
• B. $32\omega$
• C. $-32$
• D. $32$

Answer 1

The correct option is D $32$

${(1-\omega +{\omega }^2)}^5+{(1-\omega -{\omega }^2)}^5$
$={(-\omega -\omega )}^5+{(-{\omega }^2-{\omega }^2)}^5(\because 1+\omega +{\omega }^2=0)$
$={(-2\omega )}^5+{(-2{\omega }^2)}^5$
$={-2}^5({\omega }^5+{\omega }^{10})$
$=-32({\omega }^2+\omega )$
$=32$