Mathematica seems to have trouble with expressions like

$$\sqrt{\frac{1}{x}} \sqrt{x}$$

Oddly, it doesn't have trouble with


which will happily simplify to $1$ even without invoking Simplify. I know that usually FullSimplify not doing what you expect is because of unconscious assumptions you're not telling Mathematica about, but I'm at a loss as for what value of $x$ I could be talking about where the above two expressions wouldn't be equal...

At any rate, is there any way to force FullSimplify to give the expected result (i.e. $1$) in the first of the above cases?

  • 2
    $\begingroup$ Test several assumptions and you will find a counterexample: Grid[Assuming[#, {HoldForm[#], Sqrt[1/x] Sqrt[x]} // Simplify] & /@ {x > 0, x >= 0, x <= 0, x < 0}] $\endgroup$ – Bob Hanlon Oct 27 '19 at 23:48

Your requested simplification is not true for negative numbers:

Sqrt[1/x] Sqrt[x] /. x -> -1


You can give FullSimplify an assumption to get your desired output:

FullSimplify[Sqrt[1/x] Sqrt[x], x > 0]


  • $\begingroup$ .... Hm. Okay, so that does in fact work in this specific simplification, but when I tried that in my actual use case, it 'bounced' again. Lemme explore a bit and update my question. $\endgroup$ – linkhyrule5 Oct 28 '19 at 0:26
  • $\begingroup$ Okay, so, for more complicated examples with multiple variables, neither {_>0} nor (for example) {m, M}>0 will work. You have to write out in painstaking detail {m>0,M>0,...}. I'd like to do that automatically or at least with much less typing; is that new question territory or edit-my-question territory? $\endgroup$ – linkhyrule5 Oct 28 '19 at 0:29
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    $\begingroup$ @linkhyrule5 Try Thread[{m, M}>0]. Or Positive[{m, M}]. $\endgroup$ – Michael E2 Oct 28 '19 at 3:11
  • $\begingroup$ @MichaelE2 That works. Thanks! $\endgroup$ – linkhyrule5 Oct 28 '19 at 5:04

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