Math Problem Solver

The solutions are given by the formula: \[r = \left(-12 + \frac{12 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}} \cos{\left(\frac{\pi k}{6} \right)}}{5}\right) + \frac{12 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}} \sin{\left(\frac{\pi k}{6} \right)}}{5} i\] where \(k\) is an integer in \(0 \le k < 12\).

There are \(2\) real solutions.

There are \(10\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}r &= -12 + \frac{12 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}}}{5} \\&\approx 0.077208361\\r &= -12 - \frac{12 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}}}{5} \\&\approx -24.077208\end{aligned}\]

Solutions with nonzero imaginary part (\(8\) of \(10\) displayed): \[\begin{aligned}r &= -12 + \frac{6 \cdot 3^{\frac{3}{4}} \cdot 5^{\frac{5}{6}}}{5} + \frac{6 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}} i}{5} \\&\approx -1.5408308 + 6.0386042 i\\r &= -12 + \frac{6 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}}}{5} + \frac{6 \cdot 3^{\frac{3}{4}} \cdot 5^{\frac{5}{6}} i}{5} \\&\approx -5.9613958 + 10.459169 i\\r &= -12 + \frac{12 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}} i}{5} \\&= -12 + 12.077208 i\\r &= -12 - \frac{6 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}}}{5} + \frac{6 \cdot 3^{\frac{3}{4}} \cdot 5^{\frac{5}{6}} i}{5} \\&\approx -18.038604 + 10.459169 i\\r &= -12 - \frac{6 \cdot 3^{\frac{3}{4}} \cdot 5^{\frac{5}{6}}}{5} + \frac{6 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}} i}{5} \\&\approx -22.459169 + 6.0386042 i\\r &= -12 - \frac{6 \cdot 3^{\frac{3}{4}} \cdot 5^{\frac{5}{6}}}{5} - \frac{6 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}} i}{5} \\&\approx -22.459169 -6.0386042 i\\r &= -12 - \frac{6 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}}}{5} - \frac{6 \cdot 3^{\frac{3}{4}} \cdot 5^{\frac{5}{6}} i}{5} \\&\approx -18.038604 -10.459169 i\\r &= -12 - \frac{12 \cdot \sqrt[4]{3} \cdot 5^{\frac{5}{6}} i}{5} \\&= -12 -12.077208 i\end{aligned}\]

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