complex numbers - $1/sqrt{-a/b} = i sqrt{b/a}$ or $-isqrt{b/a}$?

In a book I am reading, I'm following an equation that has the line:

$$\frac{1}{\sqrt{-\frac{a}{b}}} = \sqrt{\frac{-b}{a}} = i\sqrt{\frac{b}{a}}$$

but while I was working ahead I did:

$$\frac{1}{\sqrt{ -\frac{a}{b}}} = \frac{1}{i \sqrt{\frac{a}{b}}} = -i\sqrt{\frac{b}{a}}$$

Which is correct? Both?

Answer

The second one is correct. Implicit in the assumptions in the first is using an identity like

$$\frac{1}{\sqrt x} = \sqrt{\frac 1 x}.$$

Although this is correct for $x \in \mathbb{R}^+$, it does not extend to negative or to complex numbers. There are quite a few false proofs based on the premise that $\sqrt{ab} = \sqrt a \cdot \sqrt{b}$ holds unconditionally!