How to Simplify $\sqrt {-X} \times \sqrt {-Y}$ For example : $\sqrt {-18} \times \sqrt {-12}$
Would I start by multiplying the 2 numbers under a square root, In which case the double negatives cancel out? 
$ \sqrt {-18\times-12} = \sqrt {216} = 6\sqrt {6}$
Or get the i out of the square roots in which case I get a  $i^2$?
$$ \sqrt {18}i \times \sqrt {12}i =  6\sqrt {6}i^2 = - 6\sqrt {6}$$
 A: Simply
$$i\sqrt{18}\cdot i\sqrt{12}$$
$$-1\sqrt{12\cdot 18}$$
$$-1\sqrt{216}$$
$$-6\sqrt{6}$$
ALWAYS start with $i$
A: I always emphasize to not use square roots on negative numbers! Even though one can extend the function $\sqrt\cdot$ to act like
$$\sqrt{-x}=i\sqrt x$$
for positive real numbers $x$, there is some random choice here: why $i\sqrt x$ and not $-i\sqrt x$? All arithmetic operations are absolutely invariant under an exchange $i\leftrightarrow-i$. So our choice is arbitrary and this is not good. Note that the same symmetry does not exist between $1$ and $-1$ and so it is absoloutely okay to define $\sqrt 4=2$ and not $-2$.
Yes I know, we learn that complex numbers are there to compute square roots of negative number, but this is more meant for applications like solving $x^2=-1$, which has well defined solutions $\{i,-i\}$. As I said, one can define the square root function $\sqrt\cdot$ to give $\sqrt{-1}=i$, but then we have to live with the fact that we cannot use our beautiful power rules!
