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.

Thank you very much to you ALL for kind repies in this topic, it was helpful. best wishes, Jacek

Thank you very much indeed, I have tried:

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

but it did not get me what I wanted. Can you please elaborate a bit on this:

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

Why is that ?

Cheers, Jacek

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.