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Math question


A square root divided by a square root. Find the domain. My question is this. The original problem places one square root symbol over the entire fraction: numerator and denominator. My understanding of the square root rules says that this is the same as the square root of the numerator over the square root of the denominator.

This would mean that the stuff under the square root on the numerator cannot be negative and the stuff under the square root on the denominator cannot be negative.

But, the examples I’m seeing in dds math class show that the numerator and denominator can BOTH be negative. So that you would do the division and get a positive number BEFORE taking the square root. Is this true?

Yes, that's correct-- negative-negative there leads to a positive (and real) solution.

It's not that you can't do the operations out of order there, but that you have to remember that the solution is a single, real number value because of the signs.

In an algebraic solution which includes variables, then, you'd have to "split" the problem into two solutions-- to account for the possibility that a variable has a negative value.

THEN, you'd have a solution which is fully real, and one that allows for the square root of a negative value.

What level of math, though? (I wouldn't expect imaginary/irrational solutions below geometry.)

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.

Both sides are defined if and only if x>=0 and y>0.

The RHS is defined and the LHS is undefined if and only if x<=0 and y<0

It never happens that the RHS is undefined and the LHS is defined.

Both sides are undefined if and only y=0 or (x<0 and y>0) or (x>0 and y<0).

Yes.

Well sqrt(x)/sqrt(y) and sqrt(x/y) are not the same function (they have different domains).

Generally the domain of sqrt(f(x))/sqrt(g(x)) is a subset of the domain of sqrt(f(x)/g(x)), and they are typically not equal.

As an exercise, compare the domains of
sqrt(x^2-2x+3)/sqrt(x^2+2x+3) and
sqrt((x^2-2x+3)/(x^2+2x+3)).

That's a remarkably basic error to appear in even one book. Does it *really* say that, even after you check for statements like "for now assume all variables represent positive quantities"?