1.4: Dealing with Rational Expressions

Back in Section 1.1, we defined five fraction properties we can use when manipulating fractions. They are copied here for convenience:

Properties of Fractions
Suppose $a,b,c,d$ are real numbers.

$\dfrac{a}{b}\cdot \dfrac{c}{d} = \dfrac{ac}{bd}$
$\dfrac{a}{b}\div \dfrac{c}{d} = \dfrac{a}{b}\cdot \dfrac{d}{c}$
$\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}$
$\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}$
$\dfrac{ac}{bc} = \dfrac{a}{b}$

We will now manipulate fractional expressions but ones that include variables.

Simplifying (property 5)

Property 5 says you can only cancel a factor within the context of the entire numerator/denominator.

To simplify, convert numerator/denominator to factors, then cancel.

Simplify the expression $\dfrac{x^2 - 1}{x^2 + x - 2}$.

Multiplying (property 1)

When multiplying, you should simplify first, then multiply.

Multiply and simplify: \[\dfrac{x^2 + 2x - 3}{x^2 + 8x + 16}\cdot \dfrac{3x + 12}{x - 1}\]

Dividing (property 2)

Divide and simplify: \[\dfrac{x-4}{x^2 - 4} \div \dfrac{x^2 - 3x - 4}{x^2 + 5x + 6}\]

Adding (property 3 + 4)

Adding fractions required you to find the LCD.

LCD requires factors.

You can group terms together with parentheses to create a factor.

Perform the indicated operation and simplify:

$\dfrac{3}{x-1} + \dfrac{x}{x+2}$
$\dfrac{1}{x^2 - 1} - \dfrac{2}{(x + 1)^2}$

Compound Fractions

A compound fraction is a fraction nested within another fraction, for example \[\dfrac{\dfrac{x}{y} + 1}{1 - \dfrac{y}{x}}\] When you're asked to "simplify a compound fraction," this means getting rid of the nested fraction by removing the internal denominators.

Simplify the following:

$\dfrac{\dfrac{x}{y} + 1}{1 - \dfrac{y}{x}}$
$\dfrac{\dfrac{1}{a+h} - \dfrac{1}{a}}{h}$

The second problem will appear in calculus again! Take note of the $h$'s cancelling through fraction law #5.

Rationalizing

Rationalizing means to remove square roots. There are two types of problems you will encounter: one factor or two term.

Rationalize the numerator: $\dfrac{\sqrt{4 + h} - 2}{h}$

Rationalize the denominator: $\dfrac{1}{\sqrt{x} + 1}$

Again, when you are asked to simplify a fractional expression, remember to simplify nested fractions!

Simplify: $\dfrac{\frac{1}{\sqrt{x}}}{x}$