What is the mistake? $$1=1$$
$$\Rightarrow\frac{-1}{1}=\frac{1}{-1}$$
$$\Rightarrow \sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}}$$
$$\Rightarrow\frac{i}{1}=\frac{1}{i}$$
$$\Rightarrow\frac{i}{2}=\frac{1}{2i}$$
$$\Rightarrow\frac{i}{2}+\frac{3}{2i} = \frac{1}{2i} +\frac{3}{2i}$$
$$\Rightarrow i(\frac{i}{2}+\frac{3}{2i} ) =  i(\frac{1}{2i} +\frac{3}{2i})$$
$$\Rightarrow\frac{-1}{2}=\frac{1}{2}$$
$$\Rightarrow1=2$$
What is wrong in this?

$$\sqrt{\frac{a}{b}} \neq \frac{\sqrt{a}}{\sqrt{b}} $$ in general unless both $a$ and $b$ are positive.
