Splitting powers

Question

The problem is to find $\sqrt{(-1)\cdot(-1)}$
Approach 1 - $\sqrt{(-1)\cdot (-1)} = \sqrt{(-1)^2} = -1$
Approach 2 - $\sqrt{(-1)\cdot (-1)} = \sqrt{1} = 1$

Which is correct and why?

Answer 1

When dealing with the imaginary unit, you need to be careful using properties such as:

$$ \sqrt{ a\cdot b } = \sqrt{a} \cdot \sqrt{b} $$

This holds for all $ a,b \in \mathbb{R}_+ \cup \{0\} $, but not for negative real numbers. Therefore, the wrong is the first:

$$ \sqrt{ (-1)\cdot (-1)} \neq i \cdot i $$

Answer 2

$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$

where a ≥ 0, b ≥ 0 Or a ≥ 0, b < 0

But NOT a < 0, b < 0

So applying in on a = b = -1 is invalid.