8.5 Simplify Complex Rational Expressions

Learning objectives

Learning objectives

• Simplify a complex rational expression by writing it as division

• Simplify a complex rational expression by using the LCD

Complex fractions are fractions in which the numerator or denominator contains a fraction. In Chapter 1 we simplified complex fractions like these:

In this section we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.

Here are a few complex rational expressions:

Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.

Simplify a Complex Rational Expression by Writing it as Division

We have already seen this complex rational expression earlier in this chapter.

We noted that fraction bars tell us to divide, so rewrote it as the division problem

Then we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.

This is one method to simplify rational expressions. We write it as if we were dividing two fractions.

Example 8.50

Solution

Are there any value(s) of $y$ that should not be allowed? The simplified rational expression has just a constant in the denominator. But the original complex rational expression had denominators of $y-3$ and ${y}^{2}-9$. This expression would be undefined if $y=3$ or $y=-3$.

Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.

Example 8.51

Solution

Example 8.52

Example 8.53

Solution

Simplify a Complex Rational Expression by Using the LCD

We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in Example 8.51. We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by $\frac{\mathrm{LCD}}{\mathrm{LCD}}$ we are multiplying by 1, so the value stays the same.

Example 8.54

Solution

Example 8.55

Be sure to start by factoring all the denominators so you can find the LCD.

Example 8.56

Solution

Example 8.57

Solution

Example 8.58

Solution

Section 8.5 Exercises

Section 8.5 Exercises

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

Simplify

In the following exercises, use either method.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

1. Simplify: $\frac{\frac{2}{{x}^{2}-1}}{\frac{3}{x+1}}.$

2. Simplify: $\frac{\frac{1}{{x}^{2}-7x+12}}{\frac{2}{x-4}}.$

3. Simplify: $\frac{\frac{1}{2}+\frac{2}{3}}{\frac{5}{6}+\frac{1}{12}}.$

1. Simplify: $\frac{\frac{3}{4}-\frac{1}{3}}{\frac{1}{8}+\frac{5}{6}}.$

2. Simplify: $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}.$

3. Simplify: $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{{a}^{2}}-\frac{1}{{b}^{2}}}.$

1. Simplify: $\frac{b-\frac{3b}{b+5}}{\frac{2}{b+5}+\frac{1}{b-5}}.$

2. Simplify: $\frac{1-\frac{3}{c+4}}{\frac{1}{c+4}+\frac{c}{3}}.$

3. Simplify: $\frac{\frac{1}{2}+\frac{1}{5}}{\frac{1}{10}+\frac{1}{5}}.$

1. Simplify: $\frac{\frac{1}{4}+\frac{3}{8}}{\frac{1}{2}-\frac{5}{16}}.$

2. Simplify: $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{a}{b}+\frac{b}{a}}.$

3. Simplify: $\frac{\frac{1}{{x}^{2}}-\frac{1}{{y}^{2}}}{\frac{1}{x}+\frac{1}{y}}.$

1. Simplify: $\frac{\frac{3}{x+2}}{\frac{5}{x-2}-\frac{3}{{x}^{2}-4}}.$

2. Simplify: $\frac{\frac{2}{x-7}-\frac{1}{x+7}}{\frac{6}{x+7}-\frac{1}{{x}^{2}-49}}.$

3. Simplify: $\frac{\frac{3}{{x}^{2}+7x+10}}{\frac{4}{x+2}+\frac{1}{x+5}}.$

1. Simplify: $\frac{\frac{4}{y+5}+\frac{2}{y+6}}{\frac{3y}{{y}^{2}+11y+30}}.$

2. Simplify: $\frac{\frac{x}{x+3}}{1+\frac{1}{x+3}}.$

3. Simplify: $\frac{1+\frac{1}{x-1}}{\frac{3}{x+1}}.$

1. $\frac{\frac{2a}{a+4}}{\frac{4{a}^{2}}{{a}^{2}-16}}$

2. $\frac{\frac{5}{{c}^{2}+5c-14}}{\frac{10}{c+7}}$

3. $\frac{\frac{1}{2}+\frac{5}{6}}{\frac{2}{3}+\frac{7}{9}}$

1. $\frac{\frac{2}{3}-\frac{1}{9}}{\frac{3}{4}+\frac{5}{6}}$

2. $\frac{\frac{n}{m}+\frac{1}{n}}{\frac{1}{n}-\frac{n}{m}}$

3. $\frac{\frac{1}{r}+\frac{1}{t}}{\frac{1}{{r}^{2}}-\frac{1}{{t}^{2}}}$

1. $\frac{x-\frac{2x}{x+3}}{\frac{1}{x+3}+\frac{1}{x-3}}$

2. $\frac{2-\frac{2}{a+3}}{\frac{1}{a+3}+\frac{a}{2}}$

3. $\frac{\frac{1}{3}+\frac{1}{8}}{\frac{1}{4}+\frac{1}{12}}$

1. $\frac{\frac{5}{6}+\frac{2}{9}}{\frac{7}{18}-\frac{1}{3}}$

2. $\frac{\frac{c}{d}+\frac{1}{d}}{\frac{1}{d}-\frac{d}{c}}$

3. $\frac{\frac{1}{p}+\frac{1}{q}}{\frac{1}{{p}^{2}}-\frac{1}{{q}^{2}}}$

1. $\frac{\frac{2}{x+5}}{\frac{3}{x-5}+\frac{1}{{x}^{2}-25}}$

2. $\frac{\frac{5}{{z}^{2}-64}+\frac{3}{z+8}}{\frac{1}{z+8}+\frac{2}{z-8}}$

3. $\frac{\frac{4}{{a}^{2}-2a-15}}{\frac{1}{a-5}+\frac{2}{a+3}}$

1. $\frac{\frac{5}{c+2}-\frac{3}{c+7}}{\frac{5c}{{c}^{2}+9c+14}}$

2. $\frac{2+\frac{1}{p-3}}{\frac{5}{p-3}}$

3. $\frac{\frac{m}{m+5}}{4+\frac{1}{m-5}}$

1. $\frac{\frac{3}{4}-\frac{2}{7}}{\frac{1}{2}+\frac{5}{14}}$

2. $\frac{\frac{v}{w}+\frac{1}{v}}{\frac{1}{v}-\frac{v}{w}}$

3. $\frac{\frac{2}{a+4}}{\frac{1}{{a}^{2}-16}}$

1. $\frac{\frac{3}{m}+\frac{3}{n}}{\frac{1}{{m}^{2}}-\frac{1}{{n}^{2}}}$

2. $\frac{x-\frac{3x}{x+2}}{\frac{3}{x+2}+\frac{3}{x-2}}$

3. Electronics The resistance of a circuit formed by connecting two resistors in parallel is $\frac{1}{\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}}$.

1. ⓐ Simplify the complex fraction $\frac{1}{\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}}$.

2. ⓑ Find the resistance of the circuit when ${R}_{1}=8$ and ${R}_{2}=12$.

1. In this section, you learned to simplify the complex fraction $\frac{\frac{3}{x+2}}{\frac{x}{{x}^{2}-4}}$ two ways:

rewriting it as a division problem

multiplying the numerator and denominator by the LCD

Which method do you prefer? Why?