Math Problem Solver

The solutions are given by the formula: \[x = \frac{2 \cdot 165^{\frac{2}{3}} \cos{\left(\frac{2 \pi k}{3} \right)}}{165} + \frac{2 \cdot 165^{\frac{2}{3}} \sin{\left(\frac{2 \pi k}{3} \right)}}{165} i\] where \(k\) is an integer in \(0 \le k < 3\).

There are \(2\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= \frac{2 \cdot 165^{\frac{2}{3}}}{165} \\&\approx 0.36464367\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= - \frac{165^{\frac{2}{3}}}{165} + \frac{\sqrt[6]{3} \cdot 55^{\frac{2}{3}} i}{55} \\&\approx -0.18232184 + 0.31579068 i\\x &= - \frac{165^{\frac{2}{3}}}{165} - \frac{\sqrt[6]{3} \cdot 55^{\frac{2}{3}} i}{55} \\&\approx -0.18232184 -0.31579068 i\end{aligned}\]

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