Math Problem Solver

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

There are \(0\) real solutions.

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

The are no real solutions.

Solutions with nonzero imaginary part: \[\begin{aligned}x &= 2 \cdot 2^{\frac{3}{4}} + 2 \cdot 2^{\frac{3}{4}} i \\&\approx 3.3635857 + 3.3635857 i\\x &= - 2 \cdot 2^{\frac{3}{4}} + 2 \cdot 2^{\frac{3}{4}} i \\&\approx -3.3635857 + 3.3635857 i\\x &= - 2 \cdot 2^{\frac{3}{4}} - 2 \cdot 2^{\frac{3}{4}} i \\&\approx -3.3635857 -3.3635857 i\\x &= 2 \cdot 2^{\frac{3}{4}} - 2 \cdot 2^{\frac{3}{4}} i \\&\approx 3.3635857 -3.3635857 i\end{aligned}\]

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