Step 1 :Given the equation \(\sqrt[5]{5x-5}=\sqrt[5]{6x+8}\)
Step 2 :Raise both sides of the equation to the power of 5 to remove the fifth root. This gives us the equation \((5x - 5)^5 = (6x + 8)^5\)
Step 3 :Solving this equation gives us five potential solutions: \[-13, -\frac{3501}{9302} - \frac{385\sqrt{5}}{9302} - \sqrt{-\frac{56233625}{43263602} - \frac{22418725\sqrt{5}}{43263602}}, -\frac{3501}{9302} - \frac{385\sqrt{5}}{9302} + \sqrt{-\frac{56233625}{43263602} - \frac{22418725\sqrt{5}}{43263602}}, -\frac{3501}{9302} + \frac{385\sqrt{5}}{9302} - \sqrt{-\frac{56233625}{43263602} + \frac{22418725\sqrt{5}}{43263602}}, -\frac{3501}{9302} + \frac{385\sqrt{5}}{9302} + \sqrt{-\frac{56233625}{43263602} + \frac{22418725\sqrt{5}}{43263602}}\]
Step 4 :However, the original equation only makes sense for real numbers, because we can't take the fifth root of a negative number. Therefore, we need to check which of these solutions are real.
Step 5 :The only real solution is -13.
Step 6 :So, the final solution to the equation is \(\boxed{-13}\)