z^3=-8
You asked:
Solve the equation \({z}^{3} = -8\) for the variable \(z\).
MathBot Answer:
The solutions are given by the formula: \[z = 2 \cos{\left(\frac{\pi}{3} + \frac{2 \pi k}{3} \right)} + 2 \sin{\left(\frac{\pi}{3} + \frac{2 \pi k}{3} \right)} i\] where \(k\) is an integer in \(0 \le k < 3\).
There is \(1\) real solution.There are \(2\) solutions with nonzero imaginary part.
Real solutions: \[\begin{aligned}z &= -2\end{aligned}\]
Solutions with nonzero imaginary part: \[\begin{aligned}z &= 1 + \sqrt{3} i \\&= 1 + 1.7320508 i\\z &= 1 - \sqrt{3} i \\&= 1 -1.7320508 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).