Consider an equation of the form \[x^{n}=a\] where \(a\) is a fixed number and \(n\) is an integer. Then the solution(s) of the above the equation are: \[x=\begin{cases} \sqrt[n]{a} & \text{($n$ is odd)}\\ \pm\sqrt[n]{a} & \text{($n$ is even and $a>0$)}\\ \text{no real solution} & \text{($n$ is even and $a<0$)} \end{cases}\]
Example 1. Solve \((x-5)^{4}=81\) .
Solution
\[\begin{align} x-5 & =\pm\sqrt[4]{81}\tag{take 4th root of both sides}\\ x-5 & =\pm\sqrt[4]{3^{4}}\\ x-5 & =\pm3\\ x & =5\pm3\tag{add 5 to both sides} \end{align}\] Therefore, the solutions are \(x=8\) and \(x=2\) .
Example 2. Solve \((u-1)^3=-8\) .
Solution
\[\begin{align} (u-1)^3&=(-2)^3 \\ u-1&=-2 && \text{(take 3rd root of both sides)}\\ u&=-2+1=-1. \end{align}\]