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Which cases Mathematica Simplify following expression?

FullSimplify[Sqrt[b /c] Sqrt[c], c != 0]  
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3 Answers 3

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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:

Mathematica graphics

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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]*)
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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]

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