# Simplifying following expression

Which cases Mathematica Simplify following expression?

FullSimplify[Sqrt[b /c] Sqrt[c], c != 0]


Which cases Mathematica Simplify following expression?

another option is to use Reduce to find what are the conditions. (I am assuming everything is real)

Reduce[Sqrt[b/c] Sqrt[c] == Sqrt[b], Reals]


gives

  c > 0 && b >= 0


For non real, remove the Reals above. Now the result is much more complicated as expected:

Simplify with the appropriate assumption, for example:

FullSimplify[Sqrt[b/c] Sqrt[c], c > 0]
(*Sqrt[b]*)


Or also:

FullSimplify[Sqrt[b/c] Sqrt[c], c < 0 && b > 0]
(*-Sqrt[b]*)
FullSimplify[Sqrt[b/c] Sqrt[c], c < 0 && b < 0]
(*Sqrt[b]*)
FullSimplify[Sqrt[b/c] Sqrt[c], c > 0 && b > 0]
(*Sqrt[b]*)


In the case of radicals, you can also use PowerExpand:

PowerExpand[Sqrt[b/c] Sqrt[c]]
(*Sqrt[b]*)


Refine also does the trick nicely. It's just a one-liner

Refine[Sqrt[b/c] Sqrt[c], c > 0]


Sqrt[b]

And just to cover the cases examined by @E.Chan-Lopez

Refine[Sqrt[b/c] Sqrt[c], c < 0 && b < 0]


returns

I Sqrt[-b]

and

Refine[Sqrt[b/c] Sqrt[c], c > 0 && b > 0]


Sqrt[b]