Math Problem Solver

The solutions are given by the formula: \[x = \sqrt[20]{30} \cos{\left(\frac{\pi k}{10} \right)} + \sqrt[20]{30} \sin{\left(\frac{\pi k}{10} \right)} i\] where \(k\) is an integer in \(0 \le k < 20\).

There are \(2\) real solutions.

There are \(18\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= \sqrt[20]{30} \\&\approx 1.1853758\\x &= - \sqrt[20]{30} \\&\approx -1.1853758\end{aligned}\]

Solutions with nonzero imaginary part (\(8\) of \(18\) displayed): \[\begin{aligned}x &= \sqrt[20]{30} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \sqrt[20]{30} i \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) \\&\approx 1.1273594 + 0.36630127 i\\x &= \sqrt[20]{30} \cdot \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right) + \sqrt[20]{30} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx 0.95898918 + 0.69674642 i\\x &= \sqrt[20]{30} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} + \sqrt[20]{30} i \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right) \\&\approx 0.69674642 + 0.95898918 i\\x &= \sqrt[20]{30} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) + \sqrt[20]{30} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx 0.36630127 + 1.1273594 i\\x &= \sqrt[20]{30} i \\&\approx 1.1853758 i\\x &= \sqrt[20]{30} \cdot \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right) + \sqrt[20]{30} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx -0.36630127 + 1.1273594 i\\x &= - \sqrt[20]{30} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} + \sqrt[20]{30} i \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right) \\&\approx -0.69674642 + 0.95898918 i\\x &= \sqrt[20]{30} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right) + \sqrt[20]{30} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx -0.95898918 + 0.69674642 i\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).