Find the square root of the following complex numbers: 8-15i - Vidyadip

Question

Find the square root of the following complex numbers:
$8-15\text{i}$

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Answer

Let $\text{z}=8-15\text{i}$
Then, $|\text{z}|=\sqrt{(8)^2+(-15)^2}$
$=\sqrt{64+225}$
$=\sqrt{289}$
$=17$
$\therefore\sqrt{8-15\text{i}}=\pm\Bigg\{\sqrt{\frac{17+8}{2}}-\text{i}\sqrt{\frac{17-8}{2}}\Bigg\} \ (\because\text{y}<0)$
$=\pm\Bigg\{\frac{5}{\sqrt{2}}-\text{i}\frac{3}{\sqrt{2}}\Bigg\}$
$=\pm\frac{1}{\sqrt{2}}\{5-3\text{i}\}$

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