50*(x^5)=500

asked by guest
on Sep 23, 2024 at 2:23 am



You asked:

Solve the equation \(50 \cdot {x}^{5} = 500\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \sqrt[5]{10} \cos{\left(\frac{2 \pi k}{5} \right)} + \sqrt[5]{10} \sin{\left(\frac{2 \pi k}{5} \right)} i\] where \(k\) is an integer in \(0 \le k < 5\).

There is \(1\) real solution.

There are \(4\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= \sqrt[5]{10} \\&\approx 1.5848932\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= \sqrt[5]{10} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) + \sqrt[5]{10} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx 0.48975893 + 1.507323 i\\x &= \sqrt[5]{10} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right) + \sqrt[5]{10} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx -1.2822055 + 0.93157685 i\\x &= \sqrt[5]{10} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right) - \sqrt[5]{10} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx -1.2822055 -0.93157685 i\\x &= \sqrt[5]{10} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) - \sqrt[5]{10} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx 0.48975893 -1.507323 i\end{aligned}\]

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

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