TITLE: Why is $i^3$ (the complex number "$i$") equal to $-i$ instead of $i$?
QUESTION [22 upvotes]: $$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i
$$
Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?

REPLY [7 votes]: Multiplying by $i$ in the complex plane amounts to a counterclockwise turn of $90$ degrees. Hence it is geometrically clear that $i^3 = -i$, as the following drawing shows

Regarding your equations, as remarked already, $\sqrt{a}\sqrt{b}=\sqrt{ab}$ holds only for positive real $a$ and $b$, so you cannot use it with complex numbers.

REPLY [2 votes]: Here is the Simple Answer to this question
as  $i^2=-1$, if you break $i^3$ into  $i^2\cdot i$ then you insert  $-1$ into the place of $i^2$ you will get $(-1)\cdot i =-i$
this is the simple trick behind it.