Find the square root of the following complex numbers: 1+4-3 - Vidyadip

Question

Find the square root of the following complex numbers:
$1+4\sqrt{-3}$

✓

Answer

Let $\text{z}=1+4\sqrt{-3}$
$=1+4\sqrt{3}\times\sqrt{-1} \ (\therefore \ \sqrt{-3}=\sqrt{3}\times\sqrt{-1})$
$\Rightarrow\text{z}=1+4\sqrt{ 3\text{i}}$
$\therefore|\text{z}|=\sqrt{(1)^2+(4\sqrt{3})^2}$
$=\sqrt{1+48}$
$=\sqrt{49}$
$=7$
Hence $\therefore\sqrt{1+4\sqrt{-3}}=\pm\Bigg\{\sqrt{\frac{7+1}{2}}+\text{i}\sqrt{\frac{7-1}{2}}\Bigg\} \ (\because\text{y}>0)$
$=\pm\Bigg\{\sqrt{\frac{8}{2}}-\text{i}\sqrt{\frac{6}{2}}\Bigg\}$
$=\pm\{\sqrt{4}+\text{i}\sqrt{3}\}$
$=\pm\{2+\text{i}\sqrt{3}\}$

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