7.3 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

Simplify a Complex Rational Expression by Writing it as Division

Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously 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.

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 complex rational expressions. We make sure the complex rational expression is of the form where one fraction is over one fraction. We then write it as if we were dividing two fractions.

Example 7.24

Solution

$\,\frac{\frac{6}{x-4}}{\frac{3}{{x}^{2}-16}}$

Rewrite the complex fraction as division. · $\,\frac{6}{x-4}\mathrm{÷}\frac{3}{{x}^{2}-16}$ Rewrite as the product of first times thereciprocal of the second. · $\,\frac{6}{x-4}\cdot\frac{{x}^{2}-16}{3}$ Factor. · $\,\frac{3\cdot{}2}{x-4}\cdot\frac{(x-4)(x+4)}{3}$ Multiply. · $\,\frac{3\cdot{}2(x-4)(x+4)}{3(x-4)}$ Remove common factors. · $\,\frac{3\cdot{}2(x-4)(x+4)}{3(x-4)}$ Simplify. · $\,2(x+4)$

Are there any value(s) of x that should not be allowed? The original complex rational expression had denominators of $x-4$ and ${x}^{2}-16.$ This expression would be undefined if $x=4$ or $x=-4.$

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 7.25

Solution

We follow the same procedure when the complex rational expression contains variables.

Example 7.26

We summarize the steps here.

Example 7.27

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 the LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in Example 7.25. 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 7.28

Solution

We will use the same example as in Example 7.26. Decide which method works better for you.

Example 7.29

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

Example 7.30

Solution

Be sure to factor the denominators first. Proceed carefully as the math can get messy!

Example 7.31

Solution

Example 7.32

Solution

Section 7.3 Exercises

Section 7.3 Exercises

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

In the following exercises, simplify each complex rational expression using either method.

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 the complex rational expression by writing it as division: $\frac{\frac{2}{{x}^{2}-1}}{\frac{3}{x+1}}.$

2. Simplify the complex rational expression by writing it as division: $\frac{\frac{1}{{x}^{2}-7x+12}}{\frac{2}{x-4}}.$

3. Simplify the complex rational expression by writing it as division: $\frac{\frac{1}{2}+\frac{2}{3}}{\frac{5}{6}+\frac{1}{12}}.$

1. Simplify the complex rational expression by writing it as division: $\frac{\frac{3}{4}-\frac{1}{3}}{\frac{1}{8}+\frac{5}{6}}.$

2. Simplify the complex rational expression by writing it as division: $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}.$

3. Simplify the complex rational expression by writing it as division: $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{{a}^{2}}-\frac{1}{{b}^{2}}}$.

1. Simplify the complex rational expression by writing it as division: $\frac{b-\frac{3b}{b+5}}{\frac{2}{b+5}+\frac{1}{b-5}}.$

2. Simplify the complex rational expression by writing it as division: $\frac{1-\frac{3}{c+4}}{\frac{1}{c+4}+\frac{c}{3}}.$

3. Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{2}+\frac{1}{5}}{\frac{1}{10}+\frac{1}{5}}.$

1. Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{4}+\frac{3}{8}}{\frac{1}{2}-\frac{5}{16}}.$

2. Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{a}{b}+\frac{b}{a}}.$

3. Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{{x}^{2}}-\frac{1}{{y}^{2}}}{\frac{1}{x}+\frac{1}{y}}.$

1. Simplify the complex rational expression by using the LCD: $\frac{\frac{3}{x+2}}{\frac{5}{x-2}-\frac{3}{{x}^{2}-4}}.$

2. Simplify the complex rational expression by using the LCD: $\frac{\frac{2}{x-7}-\frac{1}{x+7}}{\frac{6}{x+7}-\frac{1}{{x}^{2}-49}}.$

3. Simplify the complex rational expression by using the LCD: $\frac{\frac{3}{{x}^{2}+7x+10}}{\frac{4}{x+2}+\frac{1}{x+5}}.$

1. Simplify the complex rational expression by using the LCD: $\frac{\frac{4y}{y+5}+\frac{2}{y+6}}{\frac{3y}{{y}^{2}+11y+30}}.$

2. Simplify the complex rational expression by using the LCD: $\frac{\frac{x}{x+3}}{1+\frac{1}{x+3}}.$

3. Simplify the complex rational expression by using the LCD: $\frac{1+\frac{1}{x-1}}{\frac{3}{x+1}}.$

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

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

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

2. 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 or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?