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I want to know $\sqrt{-m}\sqrt{-n}=$? I tried in the following ways:

Way 1:$$\sqrt{-m}\sqrt{-n}=\sqrt{(-m)(-n)}=\sqrt{mn}.$$ Way 2:$$\sqrt{-m}\sqrt{-n}=\sqrt{m}i\sqrt{n}i=\sqrt{mn}i^2=-\sqrt{mn}$$

So I got two different values for the same $\sqrt{-m}\sqrt{-n}$. How can that be possible?


marked as duplicate by Saad, Lord Shark the Unknown, Watson, Cesareo, Brahadeesh Dec 1 '18 at 10:18

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


The rule $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ doesn't necessarily apply when $a < 0$ or $b < 0$. For example, $$-1 = i\cdot i = \sqrt{-1} \cdot \sqrt{-1} \neq \sqrt{(-1)\cdot(-1)} = \sqrt{1} = 1$$ Therefore, the 1st way is incorrect.


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