Math Problem Solver

The solutions are given by the formula: \[x = \sqrt[13]{12} \cos{\left(\frac{\pi}{13} + \frac{2 \pi k}{13} \right)} + \sqrt[13]{12} \sin{\left(\frac{\pi}{13} + \frac{2 \pi k}{13} \right)} i\] where \(k\) is an integer in \(0 \le k < 13\).

There are \(12\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= - \sqrt[13]{12} \\&\approx -1.210637\end{aligned}\]

Solutions with nonzero imaginary part (\(9\) of \(12\) displayed): \[\begin{aligned}x &= \sqrt[13]{12} \cos{\left(\frac{\pi}{13} \right)} + \sqrt[13]{12} i \sin{\left(\frac{\pi}{13} \right)} \\&\approx 1.1754581 + 0.2897244 i\\x &= \sqrt[13]{12} \cos{\left(\frac{3 \pi}{13} \right)} + \sqrt[13]{12} i \sin{\left(\frac{3 \pi}{13} \right)} \\&\approx 0.90617481 + 0.80280082 i\\x &= \sqrt[13]{12} \cos{\left(\frac{5 \pi}{13} \right)} + \sqrt[13]{12} i \sin{\left(\frac{5 \pi}{13} \right)} \\&\approx 0.4292978 + 1.1319653 i\\x &= - \sqrt[13]{12} \cos{\left(\frac{6 \pi}{13} \right)} + \sqrt[13]{12} i \sin{\left(\frac{6 \pi}{13} \right)} \\&\approx -0.14592616 + 1.2018101 i\\x &= - \sqrt[13]{12} \cos{\left(\frac{4 \pi}{13} \right)} + \sqrt[13]{12} i \sin{\left(\frac{4 \pi}{13} \right)} \\&\approx -0.6877202 + 0.99633472 i\\x &= - \sqrt[13]{12} \cos{\left(\frac{2 \pi}{13} \right)} + \sqrt[13]{12} i \sin{\left(\frac{2 \pi}{13} \right)} \\&\approx -1.0719658 + 0.56261107 i\\x &= - \sqrt[13]{12} \cos{\left(\frac{2 \pi}{13} \right)} - \sqrt[13]{12} i \sin{\left(\frac{2 \pi}{13} \right)} \\&\approx -1.0719658 -0.56261107 i\\x &= - \sqrt[13]{12} \cos{\left(\frac{4 \pi}{13} \right)} - \sqrt[13]{12} i \sin{\left(\frac{4 \pi}{13} \right)} \\&\approx -0.6877202 -0.99633472 i\\x &= - \sqrt[13]{12} \cos{\left(\frac{6 \pi}{13} \right)} - \sqrt[13]{12} i \sin{\left(\frac{6 \pi}{13} \right)} \\&\approx -0.14592616 -1.2018101 i\end{aligned}\]

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