Find the square root of the following complex numbers: -5+12i - Vidyadip

Question

Find the square root of the following complex numbers:
$-5+12\text{i}$

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Answer

Let $\text{z}=-5+12\text{i}$
$\Rightarrow|\text{z}|=\sqrt{(-5)^2+12^2}$
$=\sqrt{25+144}$
$=\sqrt{169}$
$=13$
$\therefore\sqrt{-5+12\text{i}}=\pm\Bigg\{\sqrt{\frac{13+(-5)}{2}}+\text{i}\sqrt{\frac{13-(-5)}{2}}\Bigg\} \ (\because\text{y}>0)$
$=\pm\Bigg\{\sqrt{\frac{8}{2}}+\text{i}\sqrt{\frac{18}{2}}\Bigg\}$
$\pm\{2+3\text{i}\}$

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