Simplifying expressions with square roots of squares

Question

I have a large number of expressions that are of the form,

f(a,b)+Sqrt[g(a,b)^2]

I understand I can use simplify and make an assumption to get either of the two solutions,

f(a,b)-g(a,b) ; f(a,b)+g(a,b).

But the issue is I have lots of these expressions with different arguments inside the square root and don't want to individually simplify them. Can I write a function that removes the square roots and outputs the solutions with their respective conditions?

Answer 1

First, let me point out that in Mathematica this f(a,b)+Sqrt[g(a,b)^2] syntactically is not correct. One needs to put square brackets, rather than round ones. After this has been removed your expression looks as:

expr=f[a, b] + Sqrt[g[a, b]^2];

I understand your explanation such that the functions like g[a,b] are reals and, therefore, their squares are always positive, right? If so, one can introduce two rules:

rule1 = Sqrt[x_^2] :> x;
rule2 = Sqrt[x_^2] :> -x;

and apply them to the whole expression, or to its parts as follows:

expr /. rule1

(*  f[a, b] + g[a, b]  *)

Of course, you have to decide which rool to apply in what case.

Have fun!