Step 1 :Given the equation \((x-5)^{2 / 3}=64\).
Step 2 :Raise both sides of the equation to the power of \(3 / 2\) to get rid of the rational exponent, resulting in \(x - 5 = 64^{3 / 2}\).
Step 3 :Solve for \(x\) by adding 5 to both sides of the equation, resulting in \(x = 64^{3 / 2} + 5\).
Step 4 :Calculate the value of \(64^{3 / 2} + 5\) to get the final answer.
Step 5 :The solution set is \(\boxed{517}\).