1000*x^90=200000 find x

asked by guest
on Nov 17, 2024 at 9:44 am



You asked:

Solve the equation \[1000 \cdot {x}^{90} = 200000\] for the unknown \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \sqrt[30]{2} \cdot \sqrt[45]{5} \cos{\left(\frac{\pi k}{45} \right)} + \sqrt[30]{2} \cdot \sqrt[45]{5} \sin{\left(\frac{\pi k}{45} \right)} i\] where \(k\) is an integer in \(0 \le k < 90\).

There are \(2\) real solutions.

There are \(88\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= \sqrt[30]{2} \cdot \sqrt[45]{5} \\&\approx 1.0606376\\x &= - \sqrt[30]{2} \cdot \sqrt[45]{5} \\&\approx -1.0606376\end{aligned}\]

Solutions with nonzero imaginary part (\(8\) of \(88\) displayed): \[\begin{aligned}x &= \sqrt[30]{2} \cdot \sqrt[45]{5} \cos{\left(\frac{\pi}{45} \right)} + \sqrt[30]{2} \cdot \sqrt[45]{5} i \sin{\left(\frac{\pi}{45} \right)} \\&\approx 1.0580539 + 0.073986336 i\\x &= \sqrt[30]{2} \cdot \sqrt[45]{5} \cos{\left(\frac{2 \pi}{45} \right)} + \sqrt[30]{2} \cdot \sqrt[45]{5} i \sin{\left(\frac{2 \pi}{45} \right)} \\&\approx 1.0503155 + 0.14761222 i\\x &= \sqrt[30]{2} \cdot \sqrt[45]{5} \left(- \frac{1}{8} + \frac{\sqrt{5}}{8} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) + \sqrt[30]{2} \cdot \sqrt[45]{5} i \left(\frac{\sqrt{3} \cdot \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) \\&\approx 1.0374601 + 0.22051895 i\\x &= \sqrt[30]{2} \cdot \sqrt[45]{5} \cos{\left(\frac{4 \pi}{45} \right)} + \sqrt[30]{2} \cdot \sqrt[45]{5} i \sin{\left(\frac{4 \pi}{45} \right)} \\&\approx 1.0195503 + 0.29235133 i\\x &= \sqrt[30]{2} \cdot \sqrt[45]{5} \cos{\left(\frac{\pi}{9} \right)} + \sqrt[30]{2} \cdot \sqrt[45]{5} i \sin{\left(\frac{\pi}{9} \right)} \\&\approx 0.99667328 + 0.36275941 i\\x &= \sqrt[30]{2} \cdot \sqrt[45]{5} \cdot \left(\frac{1}{8} + \frac{\sqrt{5}}{8} + \frac{\sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + \sqrt[30]{2} \cdot \sqrt[45]{5} i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{3} \cdot \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2}\right) \\&\approx 0.96894062 + 0.43140016 i\\x &= \sqrt[30]{2} \cdot \sqrt[45]{5} \cos{\left(\frac{7 \pi}{45} \right)} + \sqrt[30]{2} \cdot \sqrt[45]{5} i \sin{\left(\frac{7 \pi}{45} \right)} \\&\approx 0.93648737 + 0.49793917 i\\x &= \sqrt[30]{2} \cdot \sqrt[45]{5} \cos{\left(\frac{8 \pi}{45} \right)} + \sqrt[30]{2} \cdot \sqrt[45]{5} i \sin{\left(\frac{8 \pi}{45} \right)} \\&\approx 0.89947166 + 0.56205227 i\end{aligned}\]

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

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