Question

lf $Z=-1+i\sqrt{3}$ and $n$ is a positive integer not a multiple of $3$, then $Z^{2n}+2^n{Z}^n+2^{2n}$ is equal to

• A. $0$
• B. $1$
• C. $-1$
• D. $4$

Answer 1

The correct option is A $0$

We know,

$\omega =\frac{-1+\sqrt{3}i}{2}$

${\omega }^2=\frac{-1-\sqrt{3}i}{2}$

and $1+\omega +{\omega }^2=0$

Also, ${\omega }^3=1$

$\therefore Z=2\omega$

$\therefore Z^{2n}+2^n{Z}^n+2^{2n}$

$=(2\omega {)}^{2n}+2^n(2\omega {)}^n+2^{2n}$

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

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

Given, $n$ is not a multiple of $3$.

So, ${\omega }^n=\omega$ or ${\omega }^2$

$\therefore$ For both the case, we get:

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

$=2^{2n}\times 0$

$=0$