Find $$$\sqrt{- \frac{5228171817}{100000000} - i}$$$

Find $$$\sqrt{- \frac{5228171817}{100000000} - i}$$$.

The polar form of $$$- \frac{5228171817}{100000000} - i$$$ is $$$\frac{\sqrt{27343780548073081489}}{100000000} \left(\cos{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)} \right)} + i \sin{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)} \right)}\right)$$$ (for steps, see polar form calculator).

According to the De Moivre's Formula, all $$$n$$$-th roots of a complex number $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$ are given by $$$r^{\frac{1}{n}} \left(\cos{\left(\frac{\theta + 2 \pi k}{n} \right)} + i \sin{\left(\frac{\theta + 2 \pi k}{n} \right)}\right)$$$, $$$k=\overline{0..n-1}$$$.

We have that $$$r = \frac{\sqrt{27343780548073081489}}{100000000}$$$, $$$\theta = - \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}$$$, and $$$n = 2$$$.

• $$$k = 0$$$: $$$\sqrt{\frac{\sqrt{27343780548073081489}}{100000000}} \left(\cos{\left(\frac{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}\right) + 2\cdot \pi\cdot 0}{2} \right)} + i \sin{\left(\frac{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}\right) + 2\cdot \pi\cdot 0}{2} \right)}\right) = \frac{\sqrt[4]{27343780548073081489}}{10000} \left(\cos{\left(- \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} \right)} + i \sin{\left(- \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} \right)}\right) = \frac{\sqrt[4]{27343780548073081489} \sin{\left(\frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} \right)}}{10000} - \frac{\sqrt[4]{27343780548073081489} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} \right)}}{10000}$$$
• $$$k = 1$$$: $$$\sqrt{\frac{\sqrt{27343780548073081489}}{100000000}} \left(\cos{\left(\frac{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}\right) + 2\cdot \pi\cdot 1}{2} \right)} + i \sin{\left(\frac{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}\right) + 2\cdot \pi\cdot 1}{2} \right)}\right) = \frac{\sqrt[4]{27343780548073081489}}{10000} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} + \frac{\pi}{2} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} + \frac{\pi}{2} \right)}\right) = - \frac{\sqrt[4]{27343780548073081489} \sin{\left(\frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} \right)}}{10000} + \frac{\sqrt[4]{27343780548073081489} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} \right)}}{10000}$$$

$$$\sqrt{- \frac{5228171817}{100000000} - i} = \frac{\sqrt[4]{27343780548073081489} \sin{\left(\frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} \right)}}{10000} - \frac{\sqrt[4]{27343780548073081489} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} \right)}}{10000}\approx 0.069147298993848 - 7.230940431158187 i$$$A

$$$\sqrt{- \frac{5228171817}{100000000} - i} = - \frac{\sqrt[4]{27343780548073081489} \sin{\left(\frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} \right)}}{10000} + \frac{\sqrt[4]{27343780548073081489} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{2} \right)}}{10000}\approx -0.069147298993848 + 7.230940431158187 i$$$A