Try putting it in brackets instead, like this (9!)^(1/2). This makes it even more obvious to me that 9! comes first.
Besides, Mathematica supports my view. The expression
Sqrt[9!] - 9/9 // N
equals to
601.395
You are correct. You take the square root of whatever is enclosed. Since the factorial is enclosed, you have to either take the square root of 362880 or deal with the (!)^1/2 factorial of 3...which (after writing the factorial in terms of the permutations it would represent) gives your answer as well.
Look up the beta function and you'll probably see the solution of gamma(1/2) is the square root of pi. And since the gamma function is often called the "factorial" function, the beta function sometimes gets called the fractional gamma function...
Or you can treat it purely in terms of powers (like it is in the form used in 5 o'clock) and write a proof that your answer is correct--e.g., the number of n-permutations of x by choosing an element of the set and then removing it from the set a total of n times in the form x(x-1)(x-2)...(x-n+1) can be written in terms of factorials. You can use this to get a recursive relation for the set x as a factorial (x!). If you take the square root of the recursion relation, you'll get the same answer for (x!)^1/2 as you did for the set (9)^1/2=3, where 9 is the set x in the proof.
Relating powers and factorials in terms of each other is a favorite nerd playground in math land...
