y^3-4=0
You asked:
Solve the equation \({y}^{3} - 4 = 0\) for the variable \(y\).
MathBot Answer:
The solutions are given by the formula: \[y = 2^{\frac{2}{3}} \cos{\left(\frac{2 \pi k}{3} \right)} + 2^{\frac{2}{3}} \sin{\left(\frac{2 \pi k}{3} \right)} i\] 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: \[\begin{aligned}y &= 2^{\frac{2}{3}} \\&\approx 1.5874011\end{aligned}\]
Solutions with nonzero imaginary part: \[\begin{aligned}y &= - \frac{2^{\frac{2}{3}}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{2} \\&\approx -0.79370053 + 1.3747296 i\\y &= - \frac{2^{\frac{2}{3}}}{2} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{2} \\&\approx -0.79370053 -1.3747296 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).