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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

enter image description here

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.

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  • $\begingroup$ 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
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If the numbers are both negative, then it is not true.

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

But

\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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