Question

If $z=-2+2\sqrt{3}i,$ then $z^{2n}+2^{2n}{z}^n+2^{4n}$ may be equal to

• A. $3\cdot 4^{2n}$
• B. $2^{2n}$
• C. $2\cdot 4^{2n}$
• D. $0$

Answer 1

Answer: (A), (D)

The correct options are
A $3\cdot 4^{2n}$
D $0$

$\omega =\frac{-1+\sqrt{3}i}{2}$
$z=-2+2\sqrt{3}i=4\omega$

$\therefore z^{2n}+2^{2n}{z}^n+2^{4n}=(4\omega {)}^{2n}+2^{2n}(4\omega {)}^n+2^{4n}$
$=4^{2n}\cdot {\omega }^{2n}+2^{2n}\cdot 4^n\cdot {\omega }^n+2^{4n}$
$=4^{2n}[{\omega }^{2n}+{\omega }^n+1]$
$=\{\begin{array}{c}0\text{, if}n\text{is not a multiple of}3\\ 3\cdot 4^{2n}\text{, if}n\text{is a multiple of}3\end{array}$