Extracting an expression matching a pattern from a large expression

Question

I have a Mathematica expression that contains a single square root, schematically

expr = a / (b + Sqrt[c]);

where a,b

Answer 1

If you are willing to change the assignment to expr, you can do this:

expr = Hold[a / (b + Sqrt[c])];

Cases[expr, HoldPattern @ Sqrt[x_] :> x, Infinity]

The Hold in the assignment statement prevents Mathematica from applying any simplifications to the expression. In this case, Sqrt[c] gets "simplified" into Power[c,Rational[1,2]].

The HoldPattern is essential in the Cases expression to prevent the same simplification from happening to the pattern being matched.

Answer 2

Theoretically, this should work correctly:

extractSqrt = Cases[ToBoxes@#, SqrtBox@x_ :> ToExpression@x, Infinity] &;

extractSqrt[expr]

Answer 3

I await a few examples, but in the meantime, try:

Cases[expr, x_^(1/2 | -1/2) :> x, Infinity]

The standard internal form for Sqrt(x) is Power[x, 1/2].