Math Problem Solver

The solutions are given by the formula: \[x = \frac{5^{\frac{2}{5}} \cdot \sqrt[5]{147} \cos{\left(\frac{2 \pi k}{5} \right)}}{5} + \frac{5^{\frac{2}{5}} \cdot \sqrt[5]{147} \sin{\left(\frac{2 \pi k}{5} \right)}}{5} i\] where \(k\) is an integer in \(0 \le k < 5\).

There are \(4\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= \frac{5^{\frac{2}{5}} \cdot \sqrt[5]{147}}{5} \\&\approx 1.0329551\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= \frac{\sqrt[5]{147} \cdot 5^{\frac{2}{5}} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{5} + \frac{\sqrt[5]{147} \cdot 5^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{5} \\&\approx 0.3192007 + 0.98239872 i\\x &= \frac{\sqrt[5]{147} \cdot 5^{\frac{2}{5}} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right)}{5} + \frac{\sqrt[5]{147} \cdot 5^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{5} \\&\approx -0.83567827 + 0.6071558 i\\x &= \frac{\sqrt[5]{147} \cdot 5^{\frac{2}{5}} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right)}{5} - \frac{\sqrt[5]{147} \cdot 5^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{5} \\&\approx -0.83567827 -0.6071558 i\\x &= \frac{\sqrt[5]{147} \cdot 5^{\frac{2}{5}} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{5} - \frac{\sqrt[5]{147} \cdot 5^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{5} \\&\approx 0.3192007 -0.98239872 i\end{aligned}\]

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