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

Question

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

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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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