TITLE: Confused with imaginary numbers
QUESTION [6 upvotes]: In 9th grade I had an argument with my teacher that 
${i}^{3}=i$
where $i=\sqrt{-1}$
But my teacher insisted (as is the accepted case) that:
${i}^{3}=-i$
My Solution:
${i}^3=(\sqrt{-1})^3$
${i}^3=\sqrt{(-1)^3}$
${i}^3=\sqrt{-1\times-1\times-1}$
${i}^3=\sqrt{-1}$
${i}^3=i$
Generally accepted solution:
${i}^3=(\sqrt{-1})^3$
${i}^3=\sqrt{-1}\times\sqrt{-1}\times\sqrt{-1}$
${i}^3=-\sqrt{-1}$
${i}^3=-i$
What is so wrong with my approach? Is it not logical?
I am using the positive square root. There seems to be something about the order in which the power should be raised? There must be a logical reason, and I need help understanding it.

REPLY [12 votes]: The error is in saying $i=\sqrt{-1}$. The correct statement is $i^2=-1$ and we're only allowed to use this property.
Thus
$$
i^3=i^2\cdot i=(-1)\cdot i=-i
$$
Never substitute $\sqrt{-1}$ for $i$: it just leads to errors, because the standard identity $\sqrt{a}\sqrt{b}=\sqrt{ab}$ just holds for real and non negative numbers.