Dear Tigran,
I think that the problem is that the expression does not simplify the way you think it simplifies without making additional assumptions. You need to consider cases when the variables are negative or complex numbers etc.
I suppose that something similar to what you want happens when you make for example the following assumptions:
Simplify[Sqrt[(Sqrt[(a m)/h^5] Sqrt[a h m])/h], Assumptions -> {a > 0, m > 0, h > 0}]
That gives
This is different from what you expect, but I think that you might have made a mistake there in your calculation. An alternative set of assumptions could be:
Simplify[Sqrt[(Sqrt[(a m)/h^5] Sqrt[a h m])/h], Assumptions -> {a \[Element] Reals, m \[Element] Reals, h \[Element] Reals}]
or equivalently
Simplify[Sqrt[(Sqrt[(a m)/h^5] Sqrt[a h m])/h], Assumptions -> {{a, m, h} \[Element] Reals}]
both of which give
Hope this helps,
M.