Simplify The Complex Rational Expression

Learning Objectives

By the end of this department, you will be able to:

• Simplify a complex rational expression by writing it as sectionalization
• Simplify a complex rational expression by using the LCD

Be Prepared 7.7

Before you get started, take this readiness quiz.

Simplify: $\frac{\frac{3}{\mathrm{five}}}{\frac{\mathrm{ix}}{10}}.$
If yous missed this problem, review Example ane.27.

Be Prepared 7.viii

Simplify: $\frac{1-\frac{1}{3}}{4^2+4·5}.$
If you missed this problem, review Example one.31.

Exist Prepared 7.9

Solve: $\frac{1}{2x}+\frac{1}{4}=\frac{1}{8}.$
If you missed this problem, review Example 2.9.

Simplify a Complex Rational Expression by Writing it as Division

Complex fractions are fractions in which the numerator or denominator contains a fraction. Nosotros previously simplified circuitous fractions similar these:

$$\begin{array}{cccccc}\frac{\frac{3}{4}}{\frac{\mathrm{five}}{8}}\hfill & & & & & \frac{\frac{x}{2}}{\frac{xy}{\mathrm{vi}}}\hfill \end{array}$$

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

Complex Rational Expression

A circuitous rational expression is a rational expression in which the numerator and/or the denominator contains a rational expression.

Here are a few complex rational expressions:

$$\frac{\frac{4}{y-\mathrm{three}}}{\frac{8}{y^2-9}}\frac{\frac{1}{\mathrm{ten}}+\frac{\mathrm{one}}{y}}{\frac{x}{y}-\frac{y}{x}}\frac{\frac{2}{\mathrm{ten}+\mathrm{half-dozen}}}{\frac{4}{x-6}-\frac{4}{x^2-36}}$$

Think, we ever exclude values that would make any denominator zero.

We will apply 2 methods to simplify complex rational expressions.

Nosotros accept already seen this complex rational expression earlier in this affiliate.

$$\frac{\frac{6x^{\mathrm{ii}}-\mathrm{vii}x+2}{4\mathrm{ten}-8}}{\frac{2x^{\mathrm{two}}-8\mathrm{ten}+3}{x^2-\mathrm{v}x+6}}$$

We noted that fraction confined tell u.s. to divide, and then rewrote information technology every bit the segmentation problem:

$$\left(\frac{6x^2-\mathrm{vii}x+2}{\mathrm{four}\mathrm{ten}-8}\right)÷\left(\frac{2{\mathrm{ten}}^2-8x+3}{x^{\mathrm{two}}-5x+6}\right).$$

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

This is one method to simplify complex rational expressions. We brand certain the circuitous rational expression is of the form where one fraction is over one fraction. We so write it as if we were dividing 2 fractions.

Example 7.24

Simplify the circuitous rational expression by writing it equally sectionalisation: $\frac{\frac{\mathrm{six}}{x-4}}{\frac{3}{x^{\mathrm{ii}}-16}}.$

Try It 7.47

Simplify the circuitous rational expression past writing it every bit sectionalisation: $\frac{\frac{\mathrm{ii}}{x^{\mathrm{two}}-1}}{\frac{\mathrm{three}}{\mathrm{ten}+\mathrm{i}}}.$

Try It vii.48

Simplify the complex rational expression by writing it every bit division: $\frac{\frac{\mathrm{i}}{x^{\mathrm{two}}-7x+12}}{\frac{2}{\mathrm{ten}-\mathrm{iv}}}.$

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

Case 7.25

Simplify the circuitous rational expression by writing it as division: $\frac{\frac{1}{\mathrm{three}}+\frac{\mathrm{i}}{6}}{\frac{1}{2}-\frac{1}{3}}.$

Attempt It seven.49

Simplify the complex rational expression by writing it as partitioning: $\frac{\frac{\mathrm{one}}{2}+\frac{\mathrm{ii}}{3}}{\frac{5}{\mathrm{six}}+\frac{\mathrm{i}}{12}}.$

Endeavour It 7.50

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

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

Example 7.26

How to Simplify a Complex Rational Expression using Division

Simplify the complex rational expression by writing it as partitioning: $\frac{\frac{1}{\mathrm{10}}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.$

Try It 7.51

Simplify the complex rational expression by writing it as sectionalisation: $\frac{\frac{\mathrm{i}}{\mathrm{10}}+\frac{\mathrm{i}}{y}}{\frac{\mathrm{i}}{x}-\frac{1}{y}}.$

Endeavor Information technology seven.52

Simplify the circuitous rational expression by writing it as division: $\frac{\frac{\mathrm{one}}{a}+\frac{1}{b}}{\frac{\mathrm{i}}{a^2}-\frac{1}{b^2}}$ .

We summarize the steps here.

How To

Simplify a complex rational expression by writing information technology every bit division.

1. Step one. Simplify the numerator and denominator.
2. Step 2. Rewrite the complex rational expression equally a division trouble.
3. Step 3. Divide the expressions.

Example vii.27

Simplify the complex rational expression by writing it as division: $\frac{\mathrm{northward}-\frac{\mathrm{iv}n}{n+5}}{\frac{1}{n+5}+\frac{1}{\mathrm{north}-5}}.$

Try Information technology 7.53

Simplify the complex rational expression past writing information technology equally division: $\frac{b-\frac{3b}{b+\mathrm{five}}}{\frac{2}{b+\mathrm{five}}+\frac{\mathrm{one}}{b-\mathrm{v}}}.$

Try It 7.54

Simplify the complex rational expression by writing information technology equally segmentation: $\frac{\mathrm{i}-\frac{3}{c+\mathrm{iv}}}{\frac{\mathrm{ane}}{c+\mathrm{four}}+\frac{c}{\mathrm{three}}}.$

Simplify a Complex Rational Expression by Using the LCD

We "cleared" the fractions by multiplying past the LCD when we solved equations with fractions. Nosotros can employ that strategy hither to simplify complex rational expressions. We will multiply the numerator and denominator by the LCD of all the rational expressions.

Let's expect at the complex rational expression we simplified one way in Example seven.25. We will simplify information technology here by multiplying the numerator and denominator by the LCD. When we multiply by $\frac{\text{LCD}}{\text{LCD}}$ we are multiplying by 1, so the value stays the same.

Example 7.28

Simplify the complex rational expression by using the LCD: $\frac{\frac{\mathrm{ane}}{\mathrm{iii}}+\frac{\mathrm{ane}}{6}}{\frac{\mathrm{i}}{\mathrm{two}}-\frac{1}{\mathrm{iii}}}.$

Endeavor Information technology 7.55

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

Endeavour It seven.56

Simplify the complex rational expression by using the LCD: $\frac{\frac{\mathrm{i}}{4}+\frac{3}{\mathrm{viii}}}{\frac{1}{2}-\frac{\mathrm{v}}{\mathrm{sixteen}}}.$

We volition use the aforementioned example every bit in Example 7.26. Determine which method works better for you.

Case seven.29

How to Simplify a Complex Rational Expressing using the LCD

Simplify the complex rational expression by using the LCD: $\frac{\frac{\mathrm{i}}{x}+\frac{\mathrm{one}}{y}}{\frac{\mathrm{ten}}{y}-\frac{y}{\mathrm{ten}}}.$

Try It 7.57

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

Endeavour It 7.58

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

How To

Simplify a complex rational expression by using the LCD.

1. Step 1. Notice the LCD of all fractions in the complex rational expression.
2. Step 2. Multiply the numerator and denominator by the LCD.
3. Step three. Simplify the expression.

Be sure to get-go by factoring all the denominators so you tin discover the LCD.

Case seven.30

Simplify the complex rational expression past using the LCD: $\frac{\frac{2}{x+6}}{\frac{4}{x-\mathrm{half\; dozen}}-\frac{4}{x^2-36}}.$

Effort It seven.59

Simplify the circuitous rational expression by using the LCD: $\frac{\frac{\mathrm{three}}{\mathrm{ten}+2}}{\frac{5}{\mathrm{10}-\mathrm{ii}}-\frac{3}{x^{\mathrm{two}}-\mathrm{four}}}.$

Try It 7.lx

Simplify the complex rational expression by using the LCD: $\frac{\frac{2}{x-\mathrm{seven}}-\frac{1}{\mathrm{ten}+7}}{\frac{\mathrm{half\; dozen}}{x+7}-\frac{1}{x^{\mathrm{two}}-49}}.$

Be sure to gene the denominators get-go. Proceed carefully every bit the math can become messy!

Instance vii.31

Simplify the complex rational expression by using the LCD: $\frac{\frac{4}{k^2-7\mathrm{yard}+12}}{\frac{\mathrm{iii}}{k-3}-\frac{\mathrm{two}}{g-4}}.$

Endeavour It 7.61

Simplify the circuitous rational expression by using the LCD: $\frac{\frac{\mathrm{three}}{{\mathrm{10}}^2+\mathrm{vii}\mathrm{ten}+\mathrm{x}}}{\frac{4}{\mathrm{10}+\mathrm{ii}}+\frac{\mathrm{i}}{\mathrm{ten}+\mathrm{v}}}.$

Try It 7.62

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

Example seven.32

Simplify the complex rational expression past using the LCD: $\frac{\frac{y}{y+1}}{1+\frac{1}{y-1}}.$

Try It 7.63

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

Try It 7.64

Simplify the complex rational expression past using the LCD: $\frac{\mathrm{one}+\frac{1}{x-1}}{\frac{\mathrm{iii}}{x+1}}.$

Section 7.3 Exercises

Practice Makes Perfect

Simplify a Circuitous Rational Expression by Writing information technology every bit Division

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

151.

$\frac{\frac{2a}{a+4}}{\frac{4a^2}{a^{\mathrm{ii}}-16}}$

152 .

$\frac{\frac{\mathrm{iii}b}{b-5}}{\frac{b^2}{b^2-25}}$

153.

$\frac{\frac{\mathrm{five}}{c^{\mathrm{two}}+5c-14}}{\frac{10}{c+7}}$

154 .

$\frac{\frac{8}{d^2+9d+18}}{\frac{12}{d+6}}$

155.

$\frac{\frac{1}{2}+\frac{5}{6}}{\frac{\mathrm{two}}{\mathrm{iii}}+\frac{7}{9}}$

156 .

$\frac{\frac{1}{2}+\frac{3}{4}}{\frac{3}{5}+\frac{7}{10}}$

157.

$\frac{\frac{2}{\mathrm{iii}}-\frac{1}{9}}{\frac{3}{\mathrm{four}}+\frac{5}{6}}$

158 .

$\frac{\frac{1}{\mathrm{ii}}-\frac{1}{6}}{\frac{2}{3}+\frac{3}{4}}$

159.

$\frac{\frac{\mathrm{north}}{m}+\frac{1}{\mathrm{northward}}}{\frac{1}{n}-\frac{\mathrm{due\; north}}{m}}$

160 .

$\frac{\frac{\mathrm{one}}{p}+\frac{p}{q}}{\frac{q}{p}-\frac{\mathrm{i}}{q}}$

161.

$\frac{\frac{1}{r}+\frac{\mathrm{ane}}{t}}{\frac{\mathrm{one}}{r^2}-\frac{\mathrm{i}}{t^2}}$

162 .

$\frac{\frac{2}{\mathrm{five}}+\frac{2}{w}}{\frac{1}{v^{\mathrm{two}}}-\frac{\mathrm{ane}}{{\mathrm{due\; west}}^2}}$

163.

$\frac{x-\frac{2\mathrm{ten}}{x+\mathrm{three}}}{\frac{\mathrm{ane}}{x+3}+\frac{1}{x-3}}$

164 .

$\frac{y-\frac{2y}{y-4}}{\frac{2}{y-\mathrm{iv}}+\frac{\mathrm{ii}}{y+\mathrm{four}}}$

165.

$\frac{\mathrm{ii}-\frac{\mathrm{ii}}{a+\mathrm{iii}}}{\frac{1}{a+3}+\frac{a}{\mathrm{ii}}}$

166 .

$\frac{4+\frac{4}{b-5}}{\frac{1}{b-5}+\frac{b}{4}}$

Simplify a Circuitous Rational Expression past Using the LCD

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

167.

$\frac{\frac{1}{3}+\frac{1}{8}}{\frac{1}{\mathrm{four}}+\frac{\mathrm{one}}{12}}$

168 .

$\frac{\frac{1}{4}+\frac{1}{9}}{\frac{\mathrm{ane}}{6}+\frac{1}{12}}$

169.

$\frac{\frac{5}{6}+\frac{2}{\mathrm{nine}}}{\frac{\mathrm{seven}}{18}-\frac{1}{\mathrm{iii}}}$

170 .

$\frac{\frac{\mathrm{i}}{6}+\frac{\mathrm{four}}{15}}{\frac{3}{5}-\frac{\mathrm{one}}{2}}$

171.

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

172 .

$\frac{\frac{\mathrm{ane}}{\mathrm{chiliad}}+\frac{\mathrm{one\; thousand}}{\mathrm{due\; north}}}{\frac{n}{\mathrm{yard}}-\frac{\mathrm{ane}}{n}}$

173.

$\frac{\frac{1}{p}+\frac{\mathrm{i}}{q}}{\frac{1}{p^{\mathrm{two}}}-\frac{1}{q^2}}$

174 .

$\frac{\frac{2}{r}+\frac{2}{t}}{\frac{1}{r^2}-\frac{\mathrm{i}}{t^2}}$

175.

$\frac{\frac{\mathrm{ii}}{\mathrm{10}+\mathrm{v}}}{\frac{3}{x-5}+\frac{\mathrm{ane}}{{\mathrm{ten}}^2-25}}$

176 .

$\frac{\frac{5}{y-4}}{\frac{3}{y+4}+\frac{2}{y^{\mathrm{ii}}-16}}$

177.

$\frac{\frac{\mathrm{v}}{z^2-64}+\frac{\mathrm{three}}{z+\mathrm{viii}}}{\frac{1}{z+8}+\frac{2}{z-8}}$

178 .

$\frac{\frac{3}{s+\mathrm{six}}+\frac{5}{\mathrm{south}-6}}{\frac{\mathrm{one}}{s^2-36}+\frac{\mathrm{four}}{s+6}}$

179.

$\frac{\frac{\mathrm{four}}{a^2-\mathrm{two}a-15}}{\frac{\mathrm{one}}{a-5}+\frac{\mathrm{ii}}{a+\mathrm{iii}}}$

180 .

$\frac{\frac{5}{b^2-\mathrm{six}b-27}}{\frac{3}{b-9}+\frac{1}{b+3}}$

181.

$\frac{\frac{5}{c+2}-\frac{3}{c+7}}{\frac{5c}{c^{\mathrm{ii}}+9c+14}}$

182 .

$\frac{\frac{6}{d-4}-\frac{2}{d+\mathrm{seven}}}{\frac{2d}{d^2+3d-28}}$

183.

$\frac{2+\frac{\mathrm{ane}}{p-3}}{\frac{5}{p-3}}$

184 .

$\frac{\frac{\mathrm{due\; north}}{n-2}}{\mathrm{iii}+\frac{5}{\mathrm{north}-2}}$

185.

$\frac{\frac{\mathrm{thousand}}{m+5}}{4+\frac{\mathrm{ane}}{\mathrm{one\; thousand}-\mathrm{five}}}$

186 .

$\frac{7+\frac{2}{q-2}}{\frac{\mathrm{one}}{q+2}}$

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

187.

$\frac{\frac{\mathrm{three}}{4}-\frac{2}{\mathrm{vii}}}{\frac{1}{\mathrm{ii}}+\frac{\mathrm{v}}{14}}$

188 .

$\frac{\frac{v}{w}+\frac{1}{v}}{\frac{\mathrm{i}}{v}-\frac{v}{\mathrm{westward}}}$

189.

$\frac{\frac{2}{a+4}}{\frac{1}{a^2-\mathrm{sixteen}}}$

190 .

$\frac{\frac{3}{b^{\mathrm{ii}}-\mathrm{iii}b-\mathrm{xl}}}{\frac{\mathrm{v}}{b+5}-\frac{\mathrm{two}}{b-8}}$

191.

$\frac{\frac{3}{g}+\frac{3}{\mathrm{due\; north}}}{\frac{\mathrm{ane}}{{\mathrm{grand}}^{\mathrm{two}}}-\frac{1}{{\mathrm{north}}^2}}$

192 .

$\frac{\frac{2}{r-\mathrm{nine}}}{\frac{\mathrm{one}}{r+9}+\frac{3}{r^2-81}}$

193.

$\frac{\mathrm{10}-\frac{3x}{x+2}}{\frac{3}{\mathrm{ten}+2}+\frac{3}{x-\mathrm{two}}}$

194 .

$\frac{\frac{y}{y+\mathrm{three}}}{2+\frac{1}{y-3}}$

Writing Exercises

195.

In this section, you learned to simplify the complex fraction $\frac{\frac{\mathrm{three}}{x+\mathrm{two}}}{\frac{\mathrm{ten}}{x^2-\mathrm{iv}}}$ two ways: rewriting information technology as a division problem or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?

196 .

Efraim wants to start simplifying the circuitous fraction $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{a}-\frac{1}{b}}$ by cancelling the variables from the numerator and denominator, $\frac{\frac{\mathrm{ane}}{\cancel{a}}+\frac{\mathrm{i}}{\cancel{b}}}{\frac{\mathrm{one}}{\cancel{a}}-\frac{1}{\cancel{b}}}.$ Explain what is incorrect with Efraim'south plan.

Self Cheque

ⓐ Afterward completing the exercises, utilise this checklist to evaluate your mastery of the objectives of this section.

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

Simplify The Complex Rational Expression,

Source: https://openstax.org/books/intermediate-algebra-2e/pages/7-3-simplify-complex-rational-expressions

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