$a^{x}+b^{y}=c^{z}$

Step 1 ：Given the equation: \(a^{x}+b^{y}=c^{z}\)

Step 2 ：Since there are multiple variables and no specific values given, we cannot find a unique solution.

Step 3 ：However, we can create a Python function to check if the equation holds true for given values of a, b, c, x, y, and z.

Step 4 ：Final Answer: \(\boxed{\text{There is no unique solution without specific values for the variables.}}\)