Math Problem Solver

The solutions are given by the formula: \[r = \left(-1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{2 \pi k}{25} \right)}}{123}\right) + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \sin{\left(\frac{2 \pi k}{25} \right)}}{123} i\] where \(k\) is an integer in \(0 \le k < 25\).

There are \(24\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898}}{123} \\&\approx 0.082766596\end{aligned}\]

Solutions with nonzero imaginary part (\(9\) of \(24\) displayed): \[\begin{aligned}r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{2 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{2 \pi}{25} \right)}}{123} \\&\approx 0.048749492 + 0.2692731 i\\r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{4 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{4 \pi}{25} \right)}}{123} \\&\approx -0.051164399 + 0.52162679 i\\r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{6 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{6 \pi}{25} \right)}}{123} \\&\approx -0.21069712 + 0.74120474 i\\r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{8 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{8 \pi}{25} \right)}}{123} \\&\approx -0.41982465 + 0.91421007 i\\r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{123} \\&\approx -0.66540672 + 1.0297722 i\\r &= -1 + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{12 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{12 \pi}{25} \right)}}{123} \\&\approx -0.93201252 + 1.08063 i\\r &= -1 - \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{11 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{11 \pi}{25} \right)}}{123} \\&\approx -1.2028902 + 1.0635878 i\\r &= -1 - \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{9 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{9 \pi}{25} \right)}}{123} \\&\approx -1.4610196 + 0.97971651 i\\r &= -1 - \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} \cos{\left(\frac{7 \pi}{25} \right)}}{123} + \frac{123^{\frac{24}{25}} \cdot \sqrt[25]{898} i \sin{\left(\frac{7 \pi}{25} \right)}}{123} \\&\approx -1.6901814 + 0.834286 i\end{aligned}\]

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