Mathematics - Complex Number Question with Solution | TestHub

${\left(\frac{z_1}{z_2}\right)}^{50}={\left(\frac{\sqrt{3}+i\sqrt{3}}{\sqrt{3}+i}\right)}^{50}$ $={\left[{\left(\frac{\sqrt{3}(1+i)}{\sqrt{3}+i}\right)}^2\right]}^{25}={\left[\frac{3(2i)}{3-1+2\sqrt{3}i}\right]}^{25}$ $={\left(\frac{3i}{1+\sqrt{3}i}\right)}^{25}=\frac{3^{25}{i}^{25}}{{\left(-2{\omega }^2\right)}^{25}}$ $$\begin{array}{l}=-i.\omega {\left(\frac{3}{2}\right)}^{25}=-\mathrm{i}\left(\frac{-1+\sqrt{3}\mathrm{i}}{2}\right){\left(\frac{3}{2}\right)}^{25}\\ ={\left(\frac{3}{2}\right)}^{25}\left(\frac{\sqrt{3}}{2}+\frac{1}{2}\mathrm{i}\right)\end{array}$$ Hence, ${\left(\frac{z_1}{z_2}\right)}^{50}$ lies in the first quadrant as both real and imaginary parts of this number are positive.