Roots of a Complex Number Calculator

Find $$$\sqrt[4]{81 i}$$$.

The polar form of $$$81 i$$$ is $$$81 \left(\cos{\left(\frac{\pi}{2} \right)} + i \sin{\left(\frac{\pi}{2} \right)}\right)$$$ (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 $$$r = 81$$$, $$$\theta = \frac{\pi}{2}$$$, and $$$n = 4$$$.

• $$$k = 0$$$: $$$\sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 0}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 0}{4} \right)}\right) = 3 \left(\cos{\left(\frac{\pi}{8} \right)} + i \sin{\left(\frac{\pi}{8} \right)}\right) = 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$$
• $$$k = 1$$$: $$$\sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 1}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 1}{4} \right)}\right) = 3 \left(\cos{\left(\frac{5 \pi}{8} \right)} + i \sin{\left(\frac{5 \pi}{8} \right)}\right) = - 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$$
• $$$k = 2$$$: $$$\sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 2}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 2}{4} \right)}\right) = 3 \left(\cos{\left(\frac{9 \pi}{8} \right)} + i \sin{\left(\frac{9 \pi}{8} \right)}\right) = - 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$$
• $$$k = 3$$$: $$$\sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 3}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 3}{4} \right)}\right) = 3 \left(\cos{\left(\frac{13 \pi}{8} \right)} + i \sin{\left(\frac{13 \pi}{8} \right)}\right) = 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$$

$$$\sqrt[4]{81 i} = 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\approx 2.77163859753386 + 1.148050297095269 i$$$A

$$$\sqrt[4]{81 i} = - 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\approx -1.148050297095269 + 2.77163859753386 i$$$A

$$$\sqrt[4]{81 i} = - 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\approx -2.77163859753386 - 1.148050297095269 i$$$A

$$$\sqrt[4]{81 i} = 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\approx 1.148050297095269 - 2.77163859753386 i$$$A