find square root of 2-36i/2+36i

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$$\begin{gathered} \Rightarrow \frac{2+36\mathrm{i}}{2-36\mathrm{i}} \\ \mathrm{Multiply} \mathrm{and} \mathrm{divide} \mathrm{by} 2+26\mathrm{i} \\ \Rightarrow \frac{2+36\mathrm{i}}{2-36\mathrm{i}}\times \frac{2+36\mathrm{i}}{2+36\mathrm{i}} \\ \mathrm{Use} {\left(\mathrm{a}+\mathrm{b}\right)}^2={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab} \mathrm{and} {\mathrm{i}}^2={\left(\sqrt{-1}\right)}^2=-1 \mathrm{and} {\mathrm{a}}^2-{\mathrm{b}}^2=\left(\mathrm{a}-\mathrm{b}\right)\left(\mathrm{a}+\mathrm{b}\right) \\ \Rightarrow \frac{{\left(2+36\mathrm{i}\right)}^2}{\left(2+36\mathrm{i}\right)\left(2-36\mathrm{i}\right)}=\frac{4-1296+144\mathrm{i}}{{\left(2\right)}^2-{\left(36\mathrm{i}\right)}^2}=\frac{-1292+144\mathrm{i}}{4+1296}=\frac{-1292}{1300}+\frac{144}{1300}\mathrm{i} \\ \Rightarrow \frac{-1292}{1300}+\frac{144}{1300}\mathrm{i}=-0.993+0.110\mathrm{i} \\ \mathrm{So} \\ \Rightarrow \sqrt{\frac{2+36\mathrm{i}}{2-36\mathrm{i}}}=\sqrt{-0.993+0.110\mathrm{i}} \\ \mathrm{Let} \sqrt{-0.993+0.110\mathrm{i}}=\mathrm{x}+\mathrm{iy} \\ \Rightarrow -0.993+0.110\mathrm{i}={\left(\mathrm{x}+\mathrm{iy}\right)}^2 \\ \Rightarrow -0.993+0.110\mathrm{i}={\mathrm{x}}^2-{\mathrm{y}}^2+2\mathrm{ixy} \\ \mathrm{So} \mathrm{compare} \mathrm{both} \mathrm{side} \\ \mathrm{So} {\mathrm{x}}^2-{\mathrm{y}}^2=-0.993....\left(1\right) \mathrm{and} 2\mathrm{xy}=0.110 \\ \mathrm{Now} \\ \Rightarrow {\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}^2={\left({\mathrm{x}}^2-{\mathrm{y}}^2\right)}^2+4{\mathrm{x}}^2{\mathrm{y}}^2={\left(-0.993\right)}^2+{\left(0.110\right)}^2=0.98+0.0121=0.9921 \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2=0.996....\left(2\right) \\ \mathrm{Add} 1 \mathrm{and} 2 \\ \Rightarrow 2{\mathrm{x}}^2=0.003 \\ \Rightarrow \mathrm{x}=\pm 0.038 \\ \mathrm{Now} \mathrm{substract} 1 \mathrm{and} 2 \\ \Rightarrow -2{\mathrm{y}}^2=-1.989 \\ \Rightarrow \mathrm{y}=\pm 0.997 \\ \mathrm{since} 2\mathrm{xy} \mathrm{is} \mathrm{positive} \mathrm{so} \mathrm{x} \mathrm{and} \mathrm{y} \mathrm{are} \mathrm{of} \mathrm{same} \mathrm{sign} \\ \Rightarrow \pm \left(0.038+0.997\mathrm{i}\right) \\ \mathrm{So} \\ \Rightarrow \sqrt{\frac{2+36\mathrm{i}}{2-36\mathrm{i}}}=\sqrt{-0.993+0.110\mathrm{i}}=\pm \left(0.038+0.997\mathrm{i}\right) \end{gathered}$$
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