The key thing to understand is the difference between:
A) x is a square root of y
B) x = sqrt(y) (normally written with the square root sign that I can't do here!)
Any positive real number y has two real square roots, of course, one being the negation of the other. But the square root sign denotes a function which, by its definition, returns the positive square root. This is just a convention - mathematicians could have agreed that the square root sign would return the negative root, or could have invented a new kind of square root sign that would. But they didn't, so school children have to learn it as it is :-)
So it is true that sqrt(x)/sqrt(y) = sqrt(x/y) WHEN BOTH SIDES ARE DEFINED, but it isn't true without that condition.
