x^5 - 1 = 0

asked by guest
on Nov 14, 2024 at 12:51 pm



You asked:

Solve the equation \({x}^{5} - 1 = 0\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \cos{\left(\frac{2 \pi k}{5} \right)} + \sin{\left(\frac{2 \pi k}{5} \right)} i\] 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: \[\begin{aligned}x &= 1\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= - \frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx 0.30901699 + 0.95105652 i\\x &= - \frac{\sqrt{5}}{4} - \frac{1}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx -0.80901699 + 0.58778525 i\\x &= - \frac{\sqrt{5}}{4} - \frac{1}{4} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx -0.80901699 -0.58778525 i\\x &= - \frac{1}{4} + \frac{\sqrt{5}}{4} - i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx 0.30901699 -0.95105652 i\end{aligned}\]

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

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