What is $\sqrt{-x^3}$?

Question

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.

Answer 1

You will have no problems if you always use $\sqrt{x^2}=|x|$.

In particular, $\sqrt{-x^3}=|x|\sqrt{-x}$. If $x$ is negative, which it is in our problem, we can replace $|x|$ by $-x$.

In your second way of solving the problem, we get $|-x|\sqrt{-x}$, but since $-x$ is positive, $|-x|=-x$. Same (correct) answer.

Answer 2

$\sqrt {-x^3} = \sqrt {-x}^3 = \sqrt {x(-1)}^3 = \sqrt {xi^2}^3 = \sqrt{x}^3 \times \sqrt{i^2}^3 = \sqrt {x}^3 \times i^3 = -i\sqrt{x}^3 = -i\sqrt{x^3}$ $\because i^3 = -\sqrt{-1} =$ {$-i : i = \sqrt {-1}$}, 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 $i\sqrt{x^3}$.