Simplifying roots of negative numbers

[maxverywell]

In this Khan Academy video



they say that it is ok to break the square root $\sqrt{a\cdot b}$, with $a, b \in \mathbb{R}$, into the product of two square roots $\sqrt{a}\cdot \sqrt{b}$, only when: (1) both are non negative, (2) one of the two is negative and the other is possitive. I know that (1) is true by definition of the square root, but is (2) true? If (2) is true then what is the explanation for why a and b cannot be both negative?

 

[mfb]

If you work with complex numbers, for every x there are always two numbers that produce x when squared. Which one do you define as square root? You can arbitrarily pick one. That leads to rules like (1) and (2), where you decide that the square root of a positive number should be positive and the square root of a negative number should have a positive imaginary part. That decision is somewhat arbitrary, and doesn't work if you go to general complex numbers.

 

[fresh_42]

As always when it comes to complex numbers, I like to recommend this Insight:
https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/