Question

If $z=(-\sqrt{3}+\sqrt{-2})(2\sqrt{3}-i)$, then

• A. $Im\left(z\right)=\sqrt{3}(1-2\sqrt{2})$
• B. $Re\left(z\right)=(-6-\sqrt{2})$
• C. $Im\left(z\right)=\sqrt{3}(1+2\sqrt{2})$
• D. $Re\left(z\right)=(-6+\sqrt{2})$

Answer 1

Answer: (C), (D)

$Re\left(z\right)=(-6+\sqrt{2})$

We have
$z=(-\sqrt{3}+\sqrt{-2})(2\sqrt{3}-i)\Rightarrow z=(-\sqrt{3}+i\sqrt{2})(2\sqrt{3}-i)$
$\Rightarrow z=-6+i\sqrt{3}+2\sqrt{6}i-\sqrt{2}{i}^2$
$\Rightarrow z=(-6+\sqrt{2})+\sqrt{3}(1+2\sqrt{2})i.$