Find the square root of the following complex numbers: -7+24i - Vidyadip

Question

Find the square root of the following complex numbers:
$-7+24\text{i}$

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Answer

Let $\text{z}=-7+24\text{i}$
$\Rightarrow|\text{z}|=\sqrt{(-7)^2+24^2}$
$=\sqrt{49+576}$
$=\sqrt{625}$
$=25$
$\therefore\sqrt{-7-24\text{i}}=\pm\Bigg\{\sqrt{\frac{25-7}{2}}+\text{i}\sqrt{\frac{25+7}{2}}\Bigg\} \ (\because\text{y}<0)$
$=\pm\Bigg\{\sqrt{\frac{18}{2}}+\text{i}\sqrt{\frac{32}{2}}\Bigg\}$
$=\pm\{\sqrt{9}+\text{i}\sqrt{16}\}$
$=\pm\{3-4\text{i}\}$

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