Question on surds rule Through the textbook, I've been taught the rule $\frac{\sqrt a}{\sqrt b} = \sqrt\frac{a}{b}$, however I realized that if all numbers are assumed to be real, and $a<0 ,b<0$, then the rule is not true as $\frac{\sqrt{-a}}{\sqrt{-b}} = \sqrt\frac{a}{b}$, whereas in $\frac{\sqrt{-a}}{\sqrt{-b}}$, the process makes it invalid as square rooting negative numbers is impossible. So does this mean that the surds rule (multiplication of surds included) is invalid for negative numbers? Or does it mean complex numbers have to be introduced in all cases?

For all nonnegative real numbers $a$ and $b$ (with $b > 0$), we have
$$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}. $$
After that, all bets are off.  The basic problem is that the function $x \mapsto x^2$ is not injective (one-to-one) over the real numbers.  Thus to define the principal square root, which is what the surd represents, we must restrict the domain of this function to the nonnegative real numbers.  Thus if we are working with real variables, then, for example, $\sqrt{a}$ is nonsense for $a < 0$.
Okay, so real variables cause problems.  What if we work with complex variables, instead?  Unfortunately, even if we are careful, we generally cannot get the "surd rule" to work as you want.  For example, if we could extend it, then we might have
$$ 1 = \sqrt{1} = \sqrt{(-1)^2} = \sqrt{-1}^2 = i^2 = -1, $$
which is clear nonsense.  The problem again comes down to how we define the square root function, and on what domain that definition holds.  There are generally two possible choices, and these choices are more or less arbitrary.
