Question

The complex number $z$ in the second quadrant which satisfies the equation $z^3=8i$ is

• A. $i-2\sqrt{3}$
• B. $2i-\sqrt{3}$
• C. $i-\sqrt{3}$
• D. $2i-3\sqrt{3}$

Answer 1

The correct option is C $i-\sqrt{3}$

$z^3=8i$
$\Rightarrow z^3=-8i^3$
$\Rightarrow {\left(\frac{z}{-2i}\right)}^3=1$
$\Rightarrow \frac{z}{-2i}=1\text{or}\omega \text{or}{\omega }^2$
$\Rightarrow z=-2i\text{or}-2i\omega \text{or}-2i{\omega }^2$
$\Rightarrow z=-2i\text{or}-2i\left(\frac{-1+\sqrt{3}i}{2}\right)\text{or}-2i\left(\frac{-1-\sqrt{3}i}{2}\right)$
Hence, $z=-2i\text{or}i+\sqrt{3}\text{or}i-\sqrt{3}$ out of which $i-\sqrt{3}$ lies in second quadrant.