If z_ 1 = 3 +i 3 and z_ 2 = 3 +i , then the complex number ( z_ 1...

$\left(\frac{z_{1}}{z_{2}}\right)^{50}=\left(\frac{\sqrt{3}+i \sqrt{3}}{\sqrt{3}+i}\right)^{50}=$\left[\left(\frac{\sqrt{3}(1+i)}{\sqrt{3}+i}\right)^{2}\right]^{25}=\left[\frac{3(2 i)}{3-1+2 \sqrt{3} i}\right]^{25}=\left(\frac{3 i}{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.