Question

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

• A. $2^{2n}$
• B. 0, if n is multiple of 3
• C. ${3.2}^{4n}$, if n is multiple of 3
• D. none of these

Answer 1

The correct option is C ${3.2}^{4n}$, if n is multiple of 3

$z=4\left(w\right)$ where w is the cube root of unity.
$z^{2n}+2^{2n}{z}^n+2^{4n}$
$=\left(4^{2n}{w}^{2n}\right)+2^{2n}{4}^n{w}^n+2^{4n}$
$={16}^n{w}^{2n}+{16}^n\left(w^n\right)+{16}^n$
$={16}^n(w^{2n}+w^n+1)$
If $n$ is a multiple of $3$, we get
$w^{2n}+w^n+1=3$
Hence for $n$ being a multiple of $3$, we get
${16}^n\left(3\right)$
$=2^{4n}\left(3\right)$