Question

If $\omega \ne 1$is a cube root of unity, then value of the expression $(1+2\omega +2{\omega }^2{)}^{10}+(2+\omega +2{\omega }^2{)}^{10}+(2+2\omega +{\omega }^2{)}^{10}$ is equal to

• A. 3
• B. 2
• C. 1
• D. 0

Answer 1

The correct option is D 0

$(1+2\omega +2{\omega }^2{)}^{10}+(2+\omega +2{\omega }^2{)}^{10}+(2+2\omega +{\omega }^2{)}^{10}$

$=(-1{)}^{10}+(-\omega {)}^{10}+(-{\omega }^2{)}^{10}$

$=1+{\omega }^{10}+{\omega }^{20}$

$=1+\omega +{\omega }^2$

$=0$