Exercise 8 Page 673

We are asked how multiplying rational expressions is similar to multiplying numerical fractions. We are also asked how these two operations differ. Let's look at their similarities and differences separately!

Similarities

Recall that multiplying numerical fractions is equivalent to multiplying their numerators and denominators. Here, $b\ne 0$ and $d\ne 0.$ Multiplying rational expressions also means multiplying their numerators and denominators. Take a look at some examples!

Multiplication	Simplify
$\frac{x}{x-1}\cdot \frac{x+1}{x^2+2}$	$\frac{x(x+1)}{(x-1)(x^2+2)}$
$\frac{x-1}{x^2-4}\cdot \frac{x^2+6}{x-1}$	$\frac{(x-1)(x^2+6)}{(x^2-4)(x-1)}$
$\frac{x-2}{x^3+1}\cdot \frac{x^5+2}{x^2-4}$	$\frac{(x-2)(x^5+2)}{(x^3+1)(x^2-4)}$

Differences

Whenever we consider a rational expression, we have to find its excluded values. Excluded values are the values of the variable which make the denominator equal to $0.$ Let's find the excluded values of the rational expressions considered earlier.

Rational Expression	Excluded Values
$\frac{x}{x-1}$ and $\frac{x^2+6}{x-1}$	$x=1$
$\frac{x+1}{x^2+2}$	None
$\frac{x-1}{x^2-4}$ and $\frac{x^5+1}{x^2-4}$	$x=\text{-}2$ and $x=2$
$\frac{x-2}{x^3+1}$	$x=\text{-}1$

When multiplying rational expressions, we have to determine the excluded values of the resulting expression. We can do it by combining the excluded values of the original rational expressions. Let's see how it works!

Simplified Multiplication	Result	Excluded Values of the Result
$\frac{x(x+1)}{(x-1)(x^2+2)}$	$\frac{x(x+1)}{(x-1)(x^2+2)}$	$x=1$
$\frac{(x-1)(x^2+6)}{(x^2-4)(x-1)}$	$\frac{x^2+6}{x^2-4}$	$x=\text{-}2,$ $x=2,$ and $x=1$
$\frac{(x-2)(x^5+2)}{(x^3+1)(x^2-4)}$	$\frac{x^5+2}{(x^3+1)(x+2)}$	$x=\text{-}1,$ $x=\text{-}2,$ and $x=2$

This is different from multiplying numerical fractions because we do not have to worry about any excluded values. If the denominators are already non-zero numbers, their product can never become $0.$