Math Problem Solver

The solutions are given by the formula: \[x = \frac{\sqrt[4]{2} \cdot 3^{\frac{3}{4}} \cos{\left(\frac{\pi k}{2} \right)}}{3} + \frac{\sqrt[4]{2} \cdot 3^{\frac{3}{4}} \sin{\left(\frac{\pi k}{2} \right)}}{3} i\] where \(k\) is an integer in \(0 \le k < 4\).

There are \(2\) real solutions.

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

Real solutions: \[\begin{aligned}x &= \frac{\sqrt[4]{2} \cdot 3^{\frac{3}{4}}}{3} \\&\approx 0.903602\\x &= - \frac{\sqrt[4]{2} \cdot 3^{\frac{3}{4}}}{3} \\&\approx -0.903602\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= \frac{\sqrt[4]{2} \cdot 3^{\frac{3}{4}}}{3} i \\&\approx 0.903602 i\\x &= - \frac{\sqrt[4]{2} \cdot 3^{\frac{3}{4}}}{3} i \\&\approx -0.903602 i\end{aligned}\]

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