Let f(x) = \sqrt{x} and g(x) = x^2. Find f(g(f(g(f(8))))).

f(8)  =  sqrt (8)

g(f(8))= g(sqrt (8))  =  8

Each function evaluation will  "flip-flop"  back and forth between these two values

So.....f(g(f(g(f(8)))))  =     sqrt (8)