Find $$$\sqrt[4]{15625 + \frac{719413999 i}{1000000000}}$$$

Find $$$\sqrt[4]{15625 + \frac{719413999 i}{1000000000}}$$$.

The polar form of $$$15625 + \frac{719413999 i}{1000000000}$$$ is $$$\frac{\sqrt{244140625517556501957172001}}{1000000000} \left(\cos{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)}\right)$$$ (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 $$$r = \frac{\sqrt{244140625517556501957172001}}{1000000000}$$$, $$$\theta = \operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}$$$, and $$$n = 4$$$.

• $$$k = 0$$$: $$$\sqrt[4]{\frac{\sqrt{244140625517556501957172001}}{1000000000}} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} + 2\cdot \pi\cdot 0}{4} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} + 2\cdot \pi\cdot 0}{4} \right)}\right) = \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001}}{1000} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}\right) = \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} \cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000} + \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000}$$$
• $$$k = 1$$$: $$$\sqrt[4]{\frac{\sqrt{244140625517556501957172001}}{1000000000}} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} + 2\cdot \pi\cdot 1}{4} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} + 2\cdot \pi\cdot 1}{4} \right)}\right) = \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001}}{1000} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} + \frac{\pi}{2} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} + \frac{\pi}{2} \right)}\right) = - \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000} + \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000}$$$
• $$$k = 2$$$: $$$\sqrt[4]{\frac{\sqrt{244140625517556501957172001}}{1000000000}} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} + 2\cdot \pi\cdot 2}{4} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} + 2\cdot \pi\cdot 2}{4} \right)}\right) = \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001}}{1000} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} + \pi \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} + \pi \right)}\right) = - \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} \cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000} - \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000}$$$
• $$$k = 3$$$: $$$\sqrt[4]{\frac{\sqrt{244140625517556501957172001}}{1000000000}} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} + 2\cdot \pi\cdot 3}{4} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} + 2\cdot \pi\cdot 3}{4} \right)}\right) = \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001}}{1000} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} + \frac{3 \pi}{2} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} + \frac{3 \pi}{2} \right)}\right) = \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000} - \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000}$$$

$$$\sqrt[4]{15625 + \frac{719413999 i}{1000000000}} = \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} \cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000} + \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000}\approx 11.180339889720948 + 0.000128692688399 i$$$A

$$$\sqrt[4]{15625 + \frac{719413999 i}{1000000000}} = - \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000} + \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000}\approx -0.000128692688399 + 11.180339889720948 i$$$A

$$$\sqrt[4]{15625 + \frac{719413999 i}{1000000000}} = - \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} \cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000} - \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000}\approx -11.180339889720948 - 0.000128692688399 i$$$A

$$$\sqrt[4]{15625 + \frac{719413999 i}{1000000000}} = \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} \sin{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000} - \frac{10^{\frac{3}{4}} \sqrt[8]{244140625517556501957172001} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{4} \right)}}{1000}\approx 0.000128692688399 - 11.180339889720948 i$$$A