8.5E: Exercises

Practice Makes Perfect

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

Example $\PageIndex{28}$

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

Answer

$\frac{a−4}{2a}$

Example $\PageIndex{29}$

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

Example $\PageIndex{30}$

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

Answer

$\frac{1}{2(c−2)}$

Example $\PageIndex{31}$

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

Example $\PageIndex{32}$

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

Answer

$\frac{24}{26}$

Example $\PageIndex{33}$

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

Example $\PageIndex{34}$

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

Answer

$\frac{20}{57}$

Example $\PageIndex{35}$

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

Example $\PageIndex{36}$

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

Answer

$\frac{n^2+m}{m−n^2}$

Example $\PageIndex{37}$

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

Example $\PageIndex{38}$

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

Answer

$\frac{rt}{t−r}$

Example $\PageIndex{39}$

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

Example $\PageIndex{40}$

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

Answer

$\frac{(x+1)(x−3)}{2}$

Example $\PageIndex{41}$

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

Example $\PageIndex{42}$

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

Answer

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

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 $\PageIndex{44}$

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

Answer

$\frac{1}{18}$

Example $\PageIndex{45}$

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

Example $\PageIndex{46}$

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

Answer

19

Example $\PageIndex{47}$

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

Example $\PageIndex{48}$

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

Answer

$\frac{c^2+c}{c−d^2}$

Example $\PageIndex{49}$

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

Example $\PageIndex{50}$

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

Answer

$\frac{pq}{q−p}$

Example $\PageIndex{51}$

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

Example $\PageIndex{52}$

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

Answer

$\frac{2x−10}{3x+16}$

Example $\PageIndex{53}$

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

Example $\PageIndex{54}$

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

Answer

$\frac{3z−19}{3z+8}$

Example $\PageIndex{55}$

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

Example $\PageIndex{56}$

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

Answer

$\frac{4}{3a−2}$

Example $\PageIndex{57}$

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

Example $\PageIndex{58}$

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

Answer

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

Example $\PageIndex{59}$

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

Example $\PageIndex{60}$

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

Answer

$\frac{(2p−5)}{5}$

Example $\PageIndex{61}$

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

Example $\PageIndex{62}$

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

Answer

$\frac{m(m−5)}{4m^2+m−95}$

Example $\PageIndex{63}$

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

​​​​​​​Simplify

In the following exercises, use either method.

Example $\PageIndex{64}$

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

Answer

$\frac{13}{24}$

Example $\PageIndex{65}$

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

Example $\PageIndex{66}$

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

Answer

2(a−4)

Example $\PageIndex{67}$

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

Example $\PageIndex{68}$

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

Answer

$\frac{3mn}{n−m}$

Example $\PageIndex{69}$

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

Example $\PageIndex{70}$

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

Answer

$\frac{(x−1)(x−2)}{6}$

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

Answer

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?

Answer

Answers will vary.

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?