$x^{6}$ -729=0

asked by guest
on Oct 23, 2024 at 3:36 pm



You asked:

Solve the equation \({x}^{6} - 729 = 0\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = 3 \cos{\left(\frac{\pi k}{3} \right)} + 3 \sin{\left(\frac{\pi k}{3} \right)} i\] where \(k\) is an integer in \(0 \le k < 6\).

There are \(2\) real solutions.

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

Real solutions: \[\begin{aligned}x &= 3\\x &= -3\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= \frac{3}{2} + \frac{3 \sqrt{3} i}{2} \\&\approx 1.5 + 2.5980762 i\\x &= - \frac{3}{2} + \frac{3 \sqrt{3} i}{2} \\&\approx -1.5 + 2.5980762 i\\x &= - \frac{3}{2} - \frac{3 \sqrt{3} i}{2} \\&\approx -1.5 -2.5980762 i\\x &= \frac{3}{2} - \frac{3 \sqrt{3} i}{2} \\&\approx 1.5 -2.5980762 i\end{aligned}\]

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

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