Rule for multiplying rational numbers The rule for multiplying rational numbers is this:

$\space\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$

Can the rule be proven or is it meant to be taken as a given?

Edit: Where $b\neq 0$ and $d\neq 0$.
One knows that $a/b$ may be defined as the number $x$ such that $bx=a$, and that $c/d$ may be defined as the number $y$ such that $dy=c$. 

If one wants the multiplication on these objects to be associative and commutative as it is on the integers, one should ask that $ac=(bx)(dy)=(bd)(xy)$ hence that the object $xy$ fits the definition of $(ac)/(bd)$.
It is more or less taken as a given. It fits the intuition and the construction of the rational numbers (which contains the definition of this multiplication) was generalized to arbitrary commutative rings (+ the choice of a multiplicative subset). This construction is called the localization (see wikipedia).