Question

Let $z_i,i=1,2,....6$ be the roots of $z^6+z^4=2$ then ${}^6{\sum }_{i=1}|z_i{|}^4$ is equal to

• A. $4$
• B. $6$
• C. $8$
• D. $10$

Answer 1

The correct option is D $10$

Let $z^2=t\Rightarrow t^3+t^2-2=0$
$\Rightarrow (t-1)(t^2+2t+2)=0$
$\Rightarrow t=1$ or $t^2+2t+2=0$
$\Rightarrow t=\pm 1$ or $t=1\pm i$
$\Rightarrow t=1\pm i\Rightarrow |z{|}^2=\sqrt{2}\Rightarrow |z{|}^4=2$
$\Rightarrow \sum _{i=1}^6|z_i{|}^4=1+1+2+2+2+2=10$