Question

Find the complex number $\omega$ satisfying the equation $z^3=8i$ and lying in the second quadrant on the complex plane.

• A. $i+\sqrt{3}$ lies in the first quadrant.
• B. $i-\sqrt{3}$ lies in the second quadrant.
• C. $-i-\sqrt{3}$ lies in the third quadrant.
• D. $-i+\sqrt{3}$ lies in the fourth quadrant.

Answer 1

The correct option is B $i-\sqrt{3}$ lies in the second quadrant.

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