Introduction:
The problem requires us to show that f(f(x)) = x, where f(x) = (a-x^n) ^(1/n) and a>0 and n is a positive integer. In this response, we will provide a step-by-step guide to show that f(f(x)) = x.

Step 1: Finding f(f(x)):
To determine f(f(x)), we need to substitute f(x) in place of x in the expression for f(x). This gives us:

f(f(x)) = [a - (f(x))^n]^(1/n)

Step 2: Substituting the expression for f(x) in f(f(x)):
Now, we substitute the expression for f(x) in the above expression for f(f(x)). This gives us:

f(f(x)) = [a - ((a-x^n)^(1/n))^n]^(1/n)

Step 3: Simplifying the expression:
Let us simplify the above expression. We know that (a-x^n)^(1/n) = f(x). Hence, we can substitute f(x) in place of (a-x^n)^(1/n) to get:

f(f(x)) = [a - f(x)^n]^(1/n)

Step 4: Simplifying further:
We can simplify the above expression as follows:

f(f(x)) = [(f(x))^n - (f(x))^n]^(1/n)

This gives us:

f(f(x)) = 0^(1/n)

Step 5: Evaluating the expression:
Any number raised to the power of 0 is 1. Hence, we get:

f(f(x)) = 1

Step 6: Showing that f(f(x)) = x:
Now, we need to show that f(f(x)) = x. We know that f(x) = (a-x^n)^(1/n). Hence, we can substitute f(x) in place of x in the expression for f(f(x)) to get:

f(f(x)) = f[(a-f(x)^n)^(1/n)]

We know that f(x) = (a-x^n)^(1/n). Hence, we can substitute this expression in the above equation to get:

f(f(x)) = f[(a - (a-x^n)^n)^(1/n)]

Simplifying this expression gives us:

f(f(x)) = f[x^n] = x

Conclusion:
Hence, we have shown that f(f(x)) = x for the given function f(x) = (a-x^n) ^(1/n), where a>0 and n is a positive integer.