It is worth mentioning that PowerExpand assumes the variables are positive (real numbers). If the variable are not always positive, then the transformations made by PowerExpand will be invalid.

PowerExpand is more or less equivalent in this case:

Simplify[Sqrt[x^(9/2)], x > 0]
(* x^(9/4) *)

The second argument specified the assumption that x is positive (and a real number). In this way, we specify exactly which variables are positive.

Another difference is that PowerExpand expands powers and logarithms. Simplify tries to contract the expression to an equivalent expression with the smallest expression tree. Expansion and contraction are sometimes in opposition to each other.

Gianluca has shown that PowerExpand does what you want. Another way to do it is with Simplify, with its second argument restricting the base to be a positive real number:

Simplify[Sqrt[x^(9/2)], x > 0]

(* x^(9/4) *)

To reverse square rooting, you need to square the expression. In Mathematica, you can do this by using the Power function. Here's how you can go from Sqrt[x^9/2] to x^9/4:

expr1 = Sqrt[x^9/2]

This creates the expression Sqrt[x^9/2]. To reverse the square root, you square the expression:

expr2 = expr1^2

This will give you the expression x^9/2. Finally, if you want to change the exponent to 1/4, you can raise the expression to the power of 1/4:

expr3 = expr2^(1/4)

This will yield x^9/4, which is the result you desire.

For that you can use PowerExpand:

PowerExpand[Sqrt[x^(9/2)]]

Beware that in the complex plane Sqrt[x^(9/2)] is not equivalent to x^(9/4):

FindInstance[Sqrt[x^(9/2)] != x^(9/4), x]

This is the reason why Mathematica does not "simplify" Sqrt[x^(9/2)] automatically.