Math Problem Solver

The solutions are given by the formula: \[R = \frac{\sqrt[8]{2} \sqrt{5} \cdot 1799^{\frac{7}{8}} \cos{\left(\frac{\pi k}{4} \right)}}{1799} + \frac{\sqrt[8]{2} \sqrt{5} \cdot 1799^{\frac{7}{8}} \sin{\left(\frac{\pi k}{4} \right)}}{1799} i\] where \(k\) is an integer in \(0 \le k < 8\).

There are \(2\) real solutions.

There are \(6\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}R &= \frac{\sqrt[8]{2} \sqrt{5} \cdot 1799^{\frac{7}{8}}}{1799} \\&\approx 0.95550916\\R &= - \frac{\sqrt[8]{2} \sqrt{5} \cdot 1799^{\frac{7}{8}}}{1799} \\&\approx -0.95550916\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}R &= \frac{1799^{\frac{7}{8}} \cdot 2^{\frac{5}{8}} \sqrt{5}}{3598} + \frac{1799^{\frac{7}{8}} \cdot 2^{\frac{5}{8}} \sqrt{5} i}{3598} \\&\approx 0.67564701 + 0.67564701 i\\R &= \frac{1799^{\frac{7}{8}} \cdot \sqrt[8]{2} \sqrt{5}}{1799} i \\&\approx 0.95550916 i\\R &= - \frac{1799^{\frac{7}{8}} \cdot 2^{\frac{5}{8}} \sqrt{5}}{3598} + \frac{1799^{\frac{7}{8}} \cdot 2^{\frac{5}{8}} \sqrt{5} i}{3598} \\&\approx -0.67564701 + 0.67564701 i\\R &= - \frac{1799^{\frac{7}{8}} \cdot 2^{\frac{5}{8}} \sqrt{5}}{3598} - \frac{1799^{\frac{7}{8}} \cdot 2^{\frac{5}{8}} \sqrt{5} i}{3598} \\&\approx -0.67564701 -0.67564701 i\\R &= - \frac{1799^{\frac{7}{8}} \cdot \sqrt[8]{2} \sqrt{5}}{1799} i \\&\approx -0.95550916 i\\R &= \frac{1799^{\frac{7}{8}} \cdot 2^{\frac{5}{8}} \sqrt{5}}{3598} - \frac{1799^{\frac{7}{8}} \cdot 2^{\frac{5}{8}} \sqrt{5} i}{3598} \\&\approx 0.67564701 -0.67564701 i\end{aligned}\]

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