Problem in complex number multiplication

Question

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?

Answer 1

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.