Confused with imaginary numbers

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.