Does "sqrt(-42) by sqrt(-6)" mean 

"sqrt(-42) MULTIPLIED by sqrt(-6)", or

sqrt(-42) DIVIDED by sqrt(-6).

Be sure you specify which if you post again, for there is a big difference.

Either way, this gets into complex numbers when we have square roots (or any
even roots of negative numbers.

We can't just multiply under the radicals like we can when there are only
positive or zeros under root radicals.

If you meant this:



we can't multiply under those radicals.

We have to take out  first:




Now we have only positive numbers under the square radical. So now we
can multiply under the radicals.  Also 



---------------------------

If you meant:

sqrt(-42) DIVIDED by sqrt(-6).



To be on the safe side, we should not divide under the radicals if there are
negative numbers under them.




Now we have only positive numbers under the radical, so we can divide them
and get:



You may notice that if we had divided the negative numbers under the
radicals, we would have gotten the right answer, but you can't trust it
because sometimes it works and sometimes it doesn't.  [It might be 
interesting to find out when it works and when it doesn't.]

Maybe it always works for dividing but not for multiplying. I dunno. You
find out!  

Edwin