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8.5E: Exercises

  • Page ID
    30270
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    Practice Makes Perfect

    Simplify a Complex Rational Expression by Writing It as Division

    In the following exercises, simplify.

    Example \(\PageIndex{28}\)

    \(\frac{\frac{2a}{a+4}}{\frac{4a^2}{a^2−16}}\)

    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.

    The above image is four columns and three rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “simplify a complex rational expression by writing it as division.”, the next row reads “simplify a complex rational expression by using the LCD.” The remaining columns are blank.

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


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