Math Problem Solver

The solutions are given by the formula: $$x = \sqrt[10]{2} \cos{\left(\frac{\pi k}{5} \right)} + \sqrt[10]{2} \sin{\left(\frac{\pi k}{5} \right)} i$$ where $k$ is an integer in $0 \le k < 10$.

There are $2$ real solutions.

There are $8$ solutions with nonzero imaginary part.

Real solutions: $$\begin{aligned}x &= \sqrt[10]{2} \\&\approx 1.0717735\\x &= - \sqrt[10]{2} \\&\approx -1.0717735\end{aligned}$$

Solutions with nonzero imaginary part: $$\begin{aligned}x &= \sqrt[10]{2} \cdot \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right) + \sqrt[10]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx 0.86708295 + 0.62997263 i\\x &= \sqrt[10]{2} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) + \sqrt[10]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx 0.33119621 + 1.0193171 i\\x &= \sqrt[10]{2} \cdot \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right) + \sqrt[10]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx -0.33119621 + 1.0193171 i\\x &= \sqrt[10]{2} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right) + \sqrt[10]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx -0.86708295 + 0.62997263 i\\x &= \sqrt[10]{2} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right) - \sqrt[10]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx -0.86708295 -0.62997263 i\\x &= \sqrt[10]{2} \cdot \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right) - \sqrt[10]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx -0.33119621 -1.0193171 i\\x &= \sqrt[10]{2} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) - \sqrt[10]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx 0.33119621 -1.0193171 i\\x &= \sqrt[10]{2} \cdot \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right) - \sqrt[10]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx 0.86708295 -0.62997263 i\end{aligned}$$

$i$ is the imaginary unit, defined as $i^2 = -1$.