# 8.5E: Exercises

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## Practice Makes Perfect

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

##### Example $$\PageIndex{28}$$

$$\frac{\frac{2a}{a+4}}{\frac{4a^2}{a^2−16}}$$

$$\frac{a−4}{2a}$$

##### Example $$\PageIndex{29}$$

$$\frac{\frac{3b}{b−5}}{\frac{b^2}{b^2−25}}$$

##### Example $$\PageIndex{30}$$

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

$$\frac{1}{2(c−2)}$$

##### Example $$\PageIndex{31}$$

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

##### Example $$\PageIndex{32}$$

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

$$\frac{24}{26}$$

##### Example $$\PageIndex{33}$$

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

##### Example $$\PageIndex{34}$$

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

$$\frac{20}{57}$$

##### Example $$\PageIndex{35}$$

$$\frac{\frac{1}{2}−\frac{1}{6}}{\frac{2}{3}+\frac{3}{4}}$$

##### Example $$\PageIndex{36}$$

$$\frac{\frac{n}{m}+\frac{1}{n}}{\frac{1}{n}−\frac{n}{m}}$$

$$\frac{n^2+m}{m−n^2}$$

##### Example $$\PageIndex{37}$$

$$\frac{\frac{1}{p}+\frac{p}{q}}{\frac{q}{p}−\frac{1}{q}}$$

##### Example $$\PageIndex{38}$$

$$\frac{\frac{1}{r}+\frac{1}{t}}{\frac{1}{r^2}−\frac{1}{t^2}}$$

$$\frac{rt}{t−r}$$

##### Example $$\PageIndex{39}$$

$$\frac{\frac{2}{v}+\frac{2}{w}}{\frac{1}{v^2}−\frac{1}{w^2}}$$

##### Example $$\PageIndex{40}$$

$$\frac{x−\frac{2x}{x+3}}{\frac{1}{x+3}+\frac{1}{x−3}}$$

$$\frac{(x+1)(x−3)}{2}$$

##### Example $$\PageIndex{41}$$

$$\frac{y−\frac{2y}{y−4}}{\frac{2}{y−4}−\frac{2}{y+4}}$$

##### Example $$\PageIndex{42}$$

$$\frac{2−\frac{2}{a+3}}{\frac{1}{a+3}+\frac{a}{2}}$$

$$\frac{4}{a+1}$$

##### Example $$\PageIndex{43}$$

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

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

##### Example $$\PageIndex{44}$$

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

$$\frac{1}{18}$$

##### Example $$\PageIndex{45}$$

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

##### Example $$\PageIndex{46}$$

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

19

##### Example $$\PageIndex{47}$$

$$\frac{\frac{1}{6}+\frac{4}{15}}{\frac{3}{5}−\frac{1}{2}}$$

##### Example $$\PageIndex{48}$$

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

$$\frac{c^2+c}{c−d^2}$$

##### Example $$\PageIndex{49}$$

$$\frac{\frac{1}{m}+\frac{m}{n}}{\frac{n}{m}−\frac{1}{n}}$$

##### Example $$\PageIndex{50}$$

$$\frac{\frac{1}{p}+\frac{1}{q}}{\frac{1}{p^2}−\frac{1}{q^2}}$$

$$\frac{pq}{q−p}$$

##### Example $$\PageIndex{51}$$

$$\frac{\frac{2}{r}+\frac{2}{t}}{\frac{1}{r^2}−\frac{1}{t^2}}$$

##### Example $$\PageIndex{52}$$

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

$$\frac{2x−10}{3x+16}$$

##### Example $$\PageIndex{53}$$

$$\frac{\frac{5}{y−4}}{\frac{3}{y+4}+\frac{2}{y^2−16}}$$

##### Example $$\PageIndex{54}$$

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

$$\frac{3z−19}{3z+8}$$

##### Example $$\PageIndex{55}$$

$$\frac{\frac{3}{s+6}+\frac{5}{s−6}}{\frac{1}{s^2−36}+\frac{4}{s+6}}$$

##### Example $$\PageIndex{56}$$

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

$$\frac{4}{3a−2}$$

##### Example $$\PageIndex{57}$$

$$\frac{\frac{5}{b^2−6b−27}}{\frac{3}{b−9}+\frac{1}{b+3}}$$

##### Example $$\PageIndex{58}$$

$$\frac{\frac{5}{c+2}−\frac{3}{c+7}}{\frac{5c}{c^2+9c+14}}$$

$$\frac{2c+29}{5c}$$

##### Example $$\PageIndex{59}$$

$$\frac{\frac{6}{d−4}−\frac{2}{d+7}}{\frac{2d}{d^2+3d−28}}$$​​​​​​​

##### Example $$\PageIndex{60}$$

$$\frac{2+\frac{1}{p−3}}{\frac{5}{p−3}}$$

$$\frac{(2p−5)}{5}$$

##### Example $$\PageIndex{61}$$

$$\frac{\frac{n}{n−2}}{3+\frac{5}{n−2}}$$

##### Example $$\PageIndex{62}$$

$$\frac{\frac{m}{m+5}}{4+\frac{1}{m−5}}$$

$$\frac{m(m−5)}{4m^2+m−95}$$

##### Example $$\PageIndex{63}$$

$$\frac{7+\frac{2}{q−2}}{\frac{1}{q+2}}$$

​​​​​​​Simplify

In the following exercises, use either method.

##### Example $$\PageIndex{64}$$

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

$$\frac{13}{24}$$

##### Example $$\PageIndex{65}$$

$$\frac{\frac{v}{w}+\frac{1}{v}}{\frac{1}{v}−\frac{v}{w}}$$

##### Example $$\PageIndex{66}$$

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

2(a−4)

##### Example $$\PageIndex{67}$$

$$\frac{\frac{3}{b^2−3b−40}}{\frac{5}{b+5}−\frac{2}{b−8}}$$

##### Example $$\PageIndex{68}$$

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

$$\frac{3mn}{n−m}$$

##### Example $$\PageIndex{69}$$

$$\frac{\frac{2}{r−9}}{\frac{1}{r+9}+\frac{3}{r^2−81}}$$

##### Example $$\PageIndex{70}$$

$$\frac{x−\frac{3x}{x+2}}{\frac{3}{x+2}+\frac{3}{x−2}}$$

$$\frac{(x−1)(x−2)}{6}$$

##### Example $$\PageIndex{71}$$

$$\frac{\frac{y}{y+3}}{2+\frac{1}{y−3}}$$​​​​​​

## Everyday Math

##### Example $$\PageIndex{72}$$

Electronics The resistance of a circuit formed by connecting two resistors in parallel is $$\frac{1}{\frac{1}{R1}+\frac{1}{R2}}$$

1. Simplify the complex fraction $$\frac{1}{\frac{1}{R1}+\frac{1}{R2}}$$
2. Find the resistance of the circuit when R1=8 and R2=12
1. $$\frac{R1R2}{R2+R1}$$
2. $$\frac{24}{5}$$​​​​​​​
##### Example $$\PageIndex{73}$$

Ironing Lenore can do the ironing for her family’s business in hh hours. Her daughter would take h+2 hours to get the ironing done. If Lenore and her daughter work together, using 2 irons, the number of hours it would take them to do all the ironing is $$\frac{1}{\frac{1}{h}+\frac{1}{h+2}}$$

1. Simplify the complex fraction $$\frac{1}{\frac{1}{h}+\frac{1}{h+2}}$$
2. Find the number of hours it would take Lenore and her daughter, working together, to get the ironing done if h=4

## Writing Exercises

##### Example $$\PageIndex{74}$$

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?

##### Example $$\PageIndex{75}$$

Efraim wants to start simplifying the complex fraction $$\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{a}−\frac{1}{b}}$$ by cancelling the variables from the numerator and denominator. Explain what is wrong with Efraim’s plan.​​​​​​​

## Self Check

ⓐ 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?

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