y $^{-4}$ = 1/81
You asked:
Solve the equation \({y}^{-4} = \frac{1}{81}\) for the variable \(y\).
MathBot Answer:
The solutions are given by the formula: \[y = 3 \cos{\left(\frac{\pi k}{2} \right)} + 3 \sin{\left(\frac{\pi k}{2} \right)} i\] 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: \[\begin{aligned}y &= 3\\y &= -3\end{aligned}\]
Solutions with nonzero imaginary part: \[\begin{aligned}y &= 3 i\\y &= -3 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).