A simple expression that Mathematica can't simplify

Question

I'm using Mathematica 9. I encountered an expression that Mathematica couldn't simplify to a simpler form:

Simplify[(Sqrt[a b] + Sqrt[a c])^4/a^2, {a > 0, b > 0, c > 0}]

(Sqrt[a b] + Sqrt[a c])^4/a^2

FullSimplify gives the same answer. However:

Simplify[(Sqrt[a b] + Sqrt[a c])^4/a^2-(Sqrt[b] + Sqrt[c])^4, {a > 0, b > 0, c > 0}]

0

, as expected. So the simplification algorithm is imperfect? Imagine that a, b and c represent more complex expressions, is there a way to work around this problem?

Edit I'll provide the original problem with more details. I have calculated a coefficient b1 to the following:

b1=(k-l)/(k+l)

where

k==Sqrt[2 m e]/h, l==Sqrt[2 m (e - V)]/h

m, e, h, k and l are real and positive (e > V)

I need to simplify the square of b1, which can be done easily by hand.

I try

Simplify[b1^2/.{k->Sqrt[2 m e]/h, l->Sqrt[2 m (e - V)]/h},{e > V > 0, m > 0, h > 0}]

The result is quite unpleasant:

$\frac{(\sqrt{m e} - \sqrt{m (e - V)})^2}{(\sqrt{m e} + \sqrt{m (e - V)})^2}$

So I decide to "help" Mathematica a bit:

Simplify[(k-l)^4/(k^2-l^2)^2/.{k->Sqrt[2 m e]/h, l->Sqrt[2 m (e - V)]/h},{e > V > 0, m > 0, h > 0}]

It returns

$\frac{(\sqrt{m e} - \sqrt{m (e - V)})^4}{m^2 V^2}$

This is the desired form. However, Mathematica refuses to take out and simplify the m factor. Even though Expand does get rid of m correctly, it won't help much because it creates a long messy expression that can't be simplified again to the compact unexpanded form.

Answer 1

I suppose you could try Expand first and then Simplify.

Simplify[Expand[(Sqrt[a b] + Sqrt[a c])^4/a^2], a > 0]

(* (Sqrt[b] + Sqrt[c])^4 *)

Though admittedly this isn't my forte.

Worth noting is that with all of your assumptions it results in an expression with higher LeafCount than the original.

Simplify[Expand[(Sqrt[a b] + Sqrt[a c])^4/a^2], a > 0 && b > 0 && c > 0]

(* b^2 + 6 b c + c^2 + 4 b Sqrt[b c] + 4 c Sqrt[b c] *)

Also of interest is that FullSimplify gives the same result as expanding with Simplify if a > 0 is the only assumption provided.

Answer 2

Nothing can be more simple. Let us do it. This is your starting expression:

 expr = (Sqrt[m*e] - Sqrt[m*(e - V)])^2/(Sqrt[m*e] + Sqrt[
    m*(e - V)])^2;

Here is a simplification function:

mySimp[expr_] := Simplify[expr, {m > 0, e > 0, V > 0}];

Now let us take the numerator and denominator of the expression:

 num1 = Numerator[expr];
den1 = Denominator[expr];

and multiply the both by the numerator:

 den2 = den1*num1 // mySimp

(*   m^2 V^2   *)

num2 = num1^2 // mySimp

(*   (Sqrt[e m] - Sqrt[m (e - V)])^4   *)

and finally construct the ratio again:

 expr2 = num2/den2

(*    (Sqrt[e m] - Sqrt[m (e - V)])^4/(m^2 V^2)   *)

Done. Have fun!