Math Problem Solver

The solutions are given by the formula: $$x = \sqrt[13]{34} \cos{\left(\frac{2 \pi k}{13} \right)} + \sqrt[13]{34} \sin{\left(\frac{2 \pi k}{13} \right)} i$$ where $k$ is an integer in $0 \le k < 13$.

There are $12$ solutions with nonzero imaginary part.

Real solutions: $$\begin{aligned}x &= \sqrt[13]{34} \\&\approx 1.3116141\end{aligned}$$

Solutions with nonzero imaginary part ($9$ of $12$ displayed): $$\begin{aligned}x &= \sqrt[13]{34} \cos{\left(\frac{2 \pi}{13} \right)} + \sqrt[13]{34} i \sin{\left(\frac{2 \pi}{13} \right)} \\&\approx 1.1613766 + 0.60953746 i\\x &= \sqrt[13]{34} \cos{\left(\frac{4 \pi}{13} \right)} + \sqrt[13]{34} i \sin{\left(\frac{4 \pi}{13} \right)} \\&\approx 0.74508172 + 1.0794372 i\\x &= \sqrt[13]{34} \cos{\left(\frac{6 \pi}{13} \right)} + \sqrt[13]{34} i \sin{\left(\frac{6 \pi}{13} \right)} \\&\approx 0.15809761 + 1.3020509 i\\x &= - \sqrt[13]{34} \cos{\left(\frac{5 \pi}{13} \right)} + \sqrt[13]{34} i \sin{\left(\frac{5 \pi}{13} \right)} \\&\approx -0.46510476 + 1.2263805 i\\x &= - \sqrt[13]{34} \cos{\left(\frac{3 \pi}{13} \right)} + \sqrt[13]{34} i \sin{\left(\frac{3 \pi}{13} \right)} \\&\approx -0.98175724 + 0.86976102 i\\x &= - \sqrt[13]{34} \cos{\left(\frac{\pi}{13} \right)} + \sqrt[13]{34} i \sin{\left(\frac{\pi}{13} \right)} \\&\approx -1.273501 + 0.31388979 i\\x &= - \sqrt[13]{34} \cos{\left(\frac{\pi}{13} \right)} - \sqrt[13]{34} i \sin{\left(\frac{\pi}{13} \right)} \\&\approx -1.273501 -0.31388979 i\\x &= - \sqrt[13]{34} \cos{\left(\frac{3 \pi}{13} \right)} - \sqrt[13]{34} i \sin{\left(\frac{3 \pi}{13} \right)} \\&\approx -0.98175724 -0.86976102 i\\x &= - \sqrt[13]{34} \cos{\left(\frac{5 \pi}{13} \right)} - \sqrt[13]{34} i \sin{\left(\frac{5 \pi}{13} \right)} \\&\approx -0.46510476 -1.2263805 i\end{aligned}$$

$i$ is the imaginary unit, defined as $i^2 = -1$.