We know that i^3 is -i . But I am getting confused, because I thought that i can be written as √(-1) and i^3 = √(-1) × √(-1) × √(-1) = √(-1 × -1 × -1) = √( (-1)^2 × -1) = √(1× -1) = √(-1) = i ( and not -i ). Please help. Sorry I couldn't use superscript because I was using my phone.
i definitely is not \sqrt{-1}. If you like (abuse of notation) \sqrt{-1} = \pm i Using (this not correct notation) \sqrt{-1}^3 = \pm i. Much better is of course i^3 = (i*i)*i = -1*i = -i
The link given by micromass has everything you need to know and you don't need to know Euler's Formula to understand what he meant. I suggest read (not skim) the link provided by micromass.
When you got to this point: $$\sqrt{-1}\cdot\sqrt{-1}\cdot\sqrt{-1}=\sqrt{(-1)\cdot(-1)\cdot(-1)},$$ you made a mistake since [itex]\sqrt{a}\sqrt{b}=\sqrt{ab}[/itex] isn't true when [itex]a,b\lt0[/itex].