Mathematica not simplifying rational powers in complex expressions

Question

I have the following expressions and I would like to obtain a final result purely in the real domain.

Refine[Sqrt[-1] / Sqrt[a^3], Assumptions -> a<0]

Refine[Sqrt[-1] / a^(3/2), Assumptions -> a<0]

The first expression is simplified to

1 / (-a)^(3/2)

while the second one is kept as

i / a^(3/2)

How can I force the first simplification?

Answer 1

One way might be

expr = Sqrt[-1]/a^(3/2);
Assuming[a < 0, Simplify[ComplexExpand[expr]]]

PowerExpand[%]

Btw, you will not get the same exact expressions in both cases you showed.

This is because $\sqrt{a^3} = a^{3/2}$ only for $a\geq 0$ and you said that $a<0$.

ClearAll[a]
Reduce[ Sqrt[a^3] == a^(3/2) && Element[a, Reals]]