Question

If $\omega (\ne 1)$ is a cube root of unity, and $(1+\omega {)}^7=A+B\omega$. Then (A, B) equals?

• A. $(1,0)$
• B. $(-1,1)$
• C. $(0,1)$
• D. $(1,1)$

Answer 1

The correct option is D $(1,1)$

Sol${}^n\Rightarrow$ As we know,

$1+\omega +{\omega }^2=0\&{\omega }^3=1$

now, $(1+\omega {)}^7$

$=\left[\right(1+\omega {)}^3{]}^2(1+\omega )$

$=(1+{\omega }^3+3{\omega }^2+3\omega {)}^2(1+\omega )$

$=(1+{\omega }^3+3({\omega }^2+\omega \left){)}^2\right(1+\omega )$

$=(1+1+3(-1\left){)}^2\right(1+\omega )$

$=(2-3{)}^2(1+\omega )$

$=1.(1+\omega )$

$=1+\omega$

$=A+B\omega$

So, $(A,B)=(1,1)$