$8x^{9}$ = -35

asked by guest
on Sep 21, 2024 at 1:33 am



You asked:

Solve the equation \(8 \cdot {x}^{9} = -35\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} \cos{\left(\frac{\pi}{9} + \frac{2 \pi k}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} \sin{\left(\frac{\pi}{9} + \frac{2 \pi k}{9} \right)}}{2} i\] where \(k\) is an integer in \(0 \le k < 9\).

There is \(1\) real solution.

There are \(8\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= - \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35}}{2} \\&\approx -1.1782021\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} i \sin{\left(\frac{\pi}{9} \right)}}{2} \\&\approx 1.1071478 + 0.40296884 i\\x &= \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} \cdot \sqrt[9]{35} i}{4} \\&\approx 0.58910104 + 1.0203529 i\\x &= - \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} \cos{\left(\frac{4 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} i \sin{\left(\frac{4 \pi}{9} \right)}}{2} \\&\approx -0.20459264 + 1.1603025 i\\x &= - \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} \cos{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} i \sin{\left(\frac{2 \pi}{9} \right)}}{2} \\&\approx -0.90255515 + 0.7573337 i\\x &= - \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} \cos{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} i \sin{\left(\frac{2 \pi}{9} \right)}}{2} \\&\approx -0.90255515 -0.7573337 i\\x &= - \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} \cos{\left(\frac{4 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} i \sin{\left(\frac{4 \pi}{9} \right)}}{2} \\&\approx -0.20459264 -1.1603025 i\\x &= \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} \cdot \sqrt[9]{35} i}{4} \\&\approx 0.58910104 -1.0203529 i\\x &= \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot \sqrt[9]{35} i \sin{\left(\frac{\pi}{9} \right)}}{2} \\&\approx 1.1071478 -0.40296884 i\end{aligned}\]

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

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