Find the square root of the following complex numbers: 4i - Vidyadip

Question

Find the square root of the following complex numbers:
$4\text{i}$

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Answer

Let $\text{z}=4\text{i}$
Then $|\text{z}|=|4\text{i}|$
$=|4\text{i}| \ (\because|\text{z}_1\text{z}_2|=|\text{z}_1|\times|\text{z}_2|)$
$=4 \ (\because|\text{i}|=1)$
$\therefore\sqrt{4\text{i}}=\pm\Bigg\{\sqrt{\frac{4+0}{2}}+\text{i}\sqrt{\frac{4-0}{2}}\Bigg\} \ (\because\text{y}>0)$
$=\pm\{\sqrt{2}+\text{i}\sqrt{2}\}$
$=\pm\sqrt{2}(1+\text{i})$

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