$\displaystyle z = 3 + 4i = 5\,\textrm{cis}\,\arctan{\frac{4}{3}}$.
You have been told repeatedly that the magnitude is r= 5 and that the argument (or amplitude), angle, is $\displaystyle tan^{-1}(3/4)$. The polar form is either $\displaystyle 5(cos(tan^{-1}(3/4)+ i sin(tan^{-1}(3/4))$ or $\displaystyle 5 e^{itan^{-1}(3/4)}$. Your equation is $\displaystyle z^2= 5 e^{i tan^{-1}(3/4)}$. $\displaystyle z= \sqrt{5}e^{i tan^{-1}(3/4)/2}$.
However, your original question was to find the square roots of 3+ 4i and bigwave, in his first response, showed you how to do that without using the polar form.