8+ x^4=56
You asked:
Solve the equation \(8 + {x}^{4} = 56\) for the variable \(x\).
MathBot Answer:
The solutions are given by the formula: \[x = 2 \cdot \sqrt[4]{3} \cos{\left(\frac{\pi k}{2} \right)} + 2 \cdot \sqrt[4]{3} \sin{\left(\frac{\pi k}{2} \right)} i\] where \(k\) is an integer in \(0 \le k < 4\).
There are \(2\) real solutions.
There are \(2\) solutions with nonzero imaginary part.
Real solutions: \[\begin{aligned}x &= 2 \cdot \sqrt[4]{3} \\&\approx 2.632148\\x &= - 2 \cdot \sqrt[4]{3} \\&\approx -2.632148\end{aligned}\]
Solutions with nonzero imaginary part: \[\begin{aligned}x &= 2 \cdot \sqrt[4]{3} i \\&\approx 2.632148 i\\x &= - 2 \cdot \sqrt[4]{3} i \\&\approx -2.632148 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).