x⁸=5

x=?

asked by guest
on Sep 22, 2024 at 12:24 am



You asked:

Solve the equation \({x}^{8} = 5\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \sqrt[8]{5} \cos{\left(\frac{\pi k}{4} \right)} + \sqrt[8]{5} \sin{\left(\frac{\pi k}{4} \right)} i\] 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: \[\begin{aligned}x &= \sqrt[8]{5} \\&\approx 1.2228445\\x &= - \sqrt[8]{5} \\&\approx -1.2228445\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= \frac{\sqrt{2} \cdot \sqrt[8]{5}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{5} i}{2} \\&\approx 0.86468167 + 0.86468167 i\\x &= \sqrt[8]{5} i \\&\approx 1.2228445 i\\x &= - \frac{\sqrt{2} \cdot \sqrt[8]{5}}{2} + \frac{\sqrt{2} \cdot \sqrt[8]{5} i}{2} \\&\approx -0.86468167 + 0.86468167 i\\x &= - \frac{\sqrt{2} \cdot \sqrt[8]{5}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{5} i}{2} \\&\approx -0.86468167 -0.86468167 i\\x &= - \sqrt[8]{5} i \\&\approx -1.2228445 i\\x &= \frac{\sqrt{2} \cdot \sqrt[8]{5}}{2} - \frac{\sqrt{2} \cdot \sqrt[8]{5} i}{2} \\&\approx 0.86468167 -0.86468167 i\end{aligned}\]

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

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