Another square-root simplification problem

Question

In a certain sequence of computations, sometimes I'll get an expression of a form such as

Sqrt[u^4 + 4 u^6]

which it is necessary to have converted "automatically", for u > 0 to:

u^2 Sqrt[1 + 4 u^2] 

The exact square-root to be converted will not always be the same, e.g,, it might be Sqrt[u^6 + 5 u^9].

I've tried Simplify, FullSimplify, and PowerExpand, all including Assumptions -> u > 0, to no avail.

I note that, by contrast,

Simplify[Sqrt[u^2 + 5 u^3], u > 0]

does yield the desired u Sort[1 + 5 u].

How can this be done?

I'd like to avoid too special a replacement rule!

Answer 1

You can try something like this using Factor to help grease the wheels:

expr = Sqrt[u^4 + 4 u^6];
PowerExpand[Factor[expr]]
(* u^2 Sqrt[1 + 4 u^2] *)

I think this will work in any of your cases.

expr = Sqrt[u^6 + 4 u^9]    
PowerExpand[Factor[expr]]
(* u^3 Sqrt[1 + 4 u^3] *)