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Evaluate $$ \sqrt{-9}\sqrt{-4} $$ Now, I am told that $\sqrt{a}\sqrt{b}=\sqrt{ab}$, so I should be able to simply write $$ \sqrt{-9}\sqrt{-4} = \sqrt{(-9)(-4)}=\sqrt{36} = 6 $$ However, I am also told that I can simplify things like $\sqrt{-9}$ to $\sqrt{9(-1)}=\sqrt{9}\sqrt{-1}=3i$. Following this reasoning, instead of the above value of 6 I get $$ \sqrt{-9}\sqrt{-4}=\sqrt{9(-1)}\sqrt{4(-1)}=(3i)(2i)=6i^2=-6 $$ So is the answer $\pm6$? The first line of thought does not seem to be missing anything but does not include $-6$. The second line does not include the $6$. Are both simultaneously needed for completeness?

Carser
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    "I am told that $\sqrt{a}\sqrt{b}=\sqrt{ab}$". Without quantifiers, this isn't a statement. When you add the quantifiers to the given equality, then you'll either realize what's wrong or someone will readily point it out for you. – Git Gud Apr 28 '16 at 18:57
  • $\sqrt{ab}=\sqrt a\sqrt b$ is valid only when at least one of $a,b$ is non-negative. – learner Apr 28 '16 at 19:19

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