Math Problem Solver

The solutions are given by the formula: \[R = \left(-1 + \frac{\sqrt[6]{3} \cdot 10^{\frac{2}{3}} \cos{\left(\frac{\pi k}{3} \right)}}{5}\right) + \frac{\sqrt[6]{3} \cdot 10^{\frac{2}{3}} \sin{\left(\frac{\pi k}{3} \right)}}{5} i\] where \(k\) is an integer in \(0 \le k < 6\).

There are \(2\) real solutions.

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

Real solutions: \[\begin{aligned}R &= -1 + \frac{\sqrt[6]{3} \cdot 10^{\frac{2}{3}}}{5} \\&\approx 0.11485111\\R &= -1 - \frac{\sqrt[6]{3} \cdot 10^{\frac{2}{3}}}{5} \\&\approx -2.1148511\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}R &= -1 + \frac{10^{\frac{2}{3}} \cdot \sqrt[6]{3}}{10} + \frac{30^{\frac{2}{3}} i}{10} \\&\approx -0.44257444 + 0.96548938 i\\R &= -1 - \frac{10^{\frac{2}{3}} \cdot \sqrt[6]{3}}{10} + \frac{30^{\frac{2}{3}} i}{10} \\&\approx -1.5574256 + 0.96548938 i\\R &= -1 - \frac{10^{\frac{2}{3}} \cdot \sqrt[6]{3}}{10} - \frac{30^{\frac{2}{3}} i}{10} \\&\approx -1.5574256 -0.96548938 i\\R &= -1 + \frac{10^{\frac{2}{3}} \cdot \sqrt[6]{3}}{10} - \frac{30^{\frac{2}{3}} i}{10} \\&\approx -0.44257444 -0.96548938 i\end{aligned}\]

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