Thread: logarithm

from what i know, for any positive number a and b, $\displaystyle \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$
i need to know if it is true all the time, whether i could say that $\displaystyle \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$ even when i don't know for sure that a and b is always positive.
put another way, if i come across a question and my answer is $\displaystyle \frac{\sqrt{m}}{\sqrt{n}}$ where m and n are unknowns and can be any number( even a negative number), do i leave my answer as $\displaystyle \frac{\sqrt{m}}{\sqrt{n}}$? would it be correct or wrong to to simplify to $\displaystyle \sqrt{\frac{m}{n}}$?

ps ignore rationalizing for a moment >.< i just want to know if $\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ always equals $\displaystyle \sqrt{\frac{a}{b}}$