Why is it that $\displaystyle \sqrt{-1}\sqrt{-1}=-1$ instead of $\displaystyle \sqrt{(-1)(-1)}=1$? What other index laws are different when involving one or more negative numbers?

Also, according to my book, $\displaystyle (-1)^{\frac{2}{6}} \neq (-1)^{\frac{1}{3}}$, but according to my calculator, this is perfectly fine. According to it, $\displaystyle (-1)^{\frac{2}{6}} \neq ((-1)^2)^6$ and $\displaystyle (-1)^{\frac{2}{6}} \neq ((-1)^{\frac{1}{6}})^2$. This is also the case when drawing graphs on my calculator.