Simplifying a symbolic product of rational powers

Question

I'm trying to symbolically evaluate a lot of equations that can consist of up to 10 parameters. The form of these expressions is a product/quotient of rational powers of the mentioned parameters, where the exponents are rational expressions containing a unknown number p. Here is a simplified example (containing parameters Bcr and Ft):

( (Ft Bcr)^(2 p + 4) * Ft^(23 p - 4) )
/ ( Bcr^(1/(2.5 p - 4)) * Ft^( (24 p + 4)/(3 p - 1) ) )^( 3 p - 6)

I would expect, that using Simplify, FullSimplify or another, similar function I could reduce this expression to a product like this:

Ft^F[p] *Bcr^G[p]

where F[p] and G[p] are some rational functions of p. Now I understand there is a problem with considering complex numbers, multiple roots etc. However, even if I use assumptions:

$Assumptions = p > 0 && Bcr > 0 && Ft > 0

Mathematica still does not evaluate the answer neatly. A follow up question would be: why such calculations take so much time, considering this seems to be the simplest of symbolic expression manipulation - something Mathematica should excel at?

So is there a way to simplify such expressions e.g. with a function similar to FullSimplify, but which ignores problems concerning domains of variables and multiple values?

Answer 1

TraditionalForm@FullSimplify@Exp[FullSimplify@PowerExpand[
  Log[((Ft Bcr)^(2 p + 4)*Ft^(23 p - 4))/(Bcr^(1/(2.5 p - 4))*
  Ft^((24 p + 4)/(3 p - 1)))^(3 p - 6)]]]

$\text{Ft}^{p+\frac{60}{3 p-1}+36} \text{Bcr}^{-\frac{1.2}{4.\, -2.5 p}+2. p+2.8}$

Answer 2

The approach of Zviovich is fine. Just to give an alternative way of doing it:

expr = ((Ft Bcr)^(2 p + 4)*Ft^(23 p - 4))/(Bcr^(1/(2.5 p - 4))*
      Ft^((24 p + 4)/(3 p - 1)))^(3 p - 6);

Now

MapAt[Simplify[Together[#]] &, expr // PowerExpand, {{1, 2}, {2, 2}}]

(* Bcr^((4. + 0.4 p - 2. p^2)/(1.6 - 1. p)) Ft^((
 24 + 107 p + 3 p^2)/(-1 + 3 p))  *)

Done.