What is $\sqrt{-x^3}$? What is $\sqrt{-x^3}$, assuming $x \in \mathbb R$ and $x < 0$? It seems as if there are two possibilities:
$\sqrt{-x^3} = \sqrt{-x\times x \times x} = \sqrt{-x \times x^2} = x\sqrt{-x}$
$\sqrt{-x^3} = \sqrt{(-x)^3} = \sqrt{(-x)\times (-x)^2} = -x\sqrt{-x}$
But I get the feeling I'm not doing the math properly in one of them.

$\sqrt {-x^3} = \sqrt {-x}^3 = \sqrt {x(-1)}^3 = \sqrt {xi^2}^3 = \sqrt{x}^3 \times \sqrt{i^2}^3 = \sqrt {x}^3 \times i^3 = -i\sqrt{x}^3 = -i\sqrt{x^3}$
$\because i^3 = -\sqrt{-1} =$ {$-i : i = \sqrt {-1}$}, assuming we are proposing that $\sqrt {-x^3} = \sqrt {(-x)^3}$. Otherwise we would write the term as not $\sqrt {x(-1)}^3$ but instead $\sqrt {x^3(-1)}$ which is equal to $i\sqrt{x^3}$.
