This is probably a very easy question, and one regarding domains more than any particular kind of failure for mathematica to provide the answer I expect. However my question is;

Why does mathematica say sqrt[g*h] === sqrt[g] * sqrt[h] is false ?

Of course with numbers from the whole numbers, this is absolutely true

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My expectations are that this is actually something I'm misunderstanding about how Mathematica deals with variables, How Simplify works...or I'm missing something in my math knowledge I didn't consider like the possible domains Mathematica assumes.

  • $\begingroup$ Evaluate each of those expressions alone to check. Without any assumptions about g and h it won't change. And that is good because the first expression is not true in general. $\endgroup$
    – Kuba
    Jun 26 '17 at 9:26

If the numbers are both negative, then it is not true.

$$ \sqrt{\left( -3\right) \left( -4\right) }=\sqrt{12}% $$


\begin{align*} \sqrt{\left( -3\right) }\sqrt{\left( -4\right) } & =\left( i\sqrt {3}\right) \left( i\sqrt{4}\right) \\ & =i^{2}\sqrt{3}\sqrt{4}\\ & =-\sqrt{12} \end{align*}

Since $\sqrt{12}\neq-\sqrt{12}$ that is why Mathematica says False for the general case.


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