When multiplying rational expressions the thing you should pay attention to is reducing fractions and shortening expressions before you start multiplying. This will significantly shorten the time it will take for you to solve the task that has been given to you.
Example 1. Multiply and reduce $ 15ab \cdot \frac{a}{3b}$
Of course, when reducing you should make sure that all of the expressions are bounded with multiplication and that you only divide numerator with denominator. You can shorten any numerator with any denominator in the product.
This is something you already learned how to do in simplifying rational expressions, only now you have one factor more.
$ 15ab \cdot \frac{a}{3b} = \frac{15a^2b}{3b} = 5a$
Example 2. Multiply and reduce $\frac{x^2 – 1}{x} \cdot \frac{2}{x + 1}$
Now, before jumping onto the multiplication the smart thing to do is to factorize numerator with denominator. When we do this, we can easily see which expressions can be shortened.
$\frac{(x – 1)(x + 1)}{x} \cdot \frac{2}{x + 1}$
We can see that the first factor and the second factor have expression $ x + 1$. Everything is bounded by multiplication so we can shorten expressions $x+1$ in each expressions.
$\frac{(x – 1)(x + 1)}{x} \cdot \frac{2}{x + 1} = \frac{2(x – 1)}{x}$
Since the numerator and denominator of this fraction have no common terms or expressions there is nothing else we can do.
Example 3. Multiply and reduce $\frac{x – 1}{5x} \cdot \frac{10}{1 – x}$
As we can notice terms in expressions $ x – 1$ and $ x + 1$ are different only in their sign. We can’t shorten them just yet, because signs make a big difference, but we can extract $-1$ from one expression to get another.
$\frac{- (- x + 1)}{5x} \cdot \frac{10}{1 – x} = – \frac{2}{x}$
Dividing rational expressions is the same as multiplying the first expression with reciprocal other.
Example 4. Divide and reduce $\frac{3x^4b}{6x^2} : \frac{b}{2}$
Reciprocal of the expression $\frac{b}{2}$ is expression $\frac{2}{b}$, which means that:
$\frac{3x^2b}{6} : \frac{b}{2} = \frac{x^2b}{2} \cdot \frac{2}{b} = \frac{x^2}{b}$
Example 5. Divide and reduce $\frac{a^2 – 9}{6a} : \frac{a – 3}{18a^3}$.
$\frac{a^2 – 9}{6a} \cdot \frac{18a^3}{a – 3} = \frac{(a – 3)(a + 3)}{6a} \cdot \frac{18a^3}{a – 3} = (a + 3)3a^2$
