Find $$$\sqrt{32 + 4 \sqrt{17} i}$$$

Find $$$\sqrt{32 + 4 \sqrt{17} i}$$$.

The polar form of $$$32 + 4 \sqrt{17} i$$$ is $$$36 \left(\cos{\left(\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} \right)}\right)$$$ (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 $$$r = 36$$$, $$$\theta = \operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}$$$, and $$$n = 2$$$.

• $$$k = 0$$$: $$$\sqrt{36} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} + 2\cdot \pi\cdot 0}{2} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} + 2\cdot \pi\cdot 0}{2} \right)}\right) = 6 \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)}\right) = 6 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)} + 6 i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)}$$$
• $$$k = 1$$$: $$$\sqrt{36} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} + 2\cdot \pi\cdot 1}{2} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} + 2\cdot \pi\cdot 1}{2} \right)}\right) = 6 \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} + \pi \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} + \pi \right)}\right) = - 6 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)} - 6 i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)}$$$

$$$\sqrt{32 + 4 \sqrt{17} i} = 6 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)} + 6 i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)}\approx 5.8309518948453 + 1.414213562373095 i$$$A

$$$\sqrt{32 + 4 \sqrt{17} i} = - 6 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)} - 6 i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)}\approx -5.8309518948453 - 1.414213562373095 i$$$A