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8.2E: Exersices

  • Page ID
    30267
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    Practice Makes Perfect

    Multiply Rational Expressions

    In the following exercises, multiply.

    Example \(\PageIndex{40}\)

    \(\frac{12}{16}·\frac{4}{10}\)

    Answer

    \(\frac{3}{10}\)

    Example \(\PageIndex{41}\)

    \(\frac{32}{5}·\frac{16}{24}\)

    Example \(\PageIndex{42}\)

    \(\frac{18}{10}·\frac{4}{30}\)

    Answer

    \(\frac{6}{25}\)

    Example \(\PageIndex{43}\)

    \(\frac{21}{36}·\frac{45}{24}\)

    Example \(\PageIndex{44}\)

    \(\frac{5x^{2}y^{4}}{12xy^3}·\frac{6x^2}{20y^2}\)

    Answer

    \(\frac{x^3}{8y}\)

    Example \(\PageIndex{45}\)

    \(\frac{8w^{3}y^9}{y^2}·\frac{3y}{4w^4}\)

    Example \(\PageIndex{46}\)

    \(\frac{12a^{3}b}{b^2}·\frac{2ab^2}{9b^3}\)

    Answer

    \(\frac{8a^4}{3b^2}\)

    Example \(\PageIndex{47}\)

    \(\frac{4mn^2}{5n^3}·\frac{mn^3}{8m^2}\)

    Example \(\PageIndex{48}\)

    \(\frac{5p^2}{p^2−5p−36}·\frac{p^2−16}{10p}\)

    Answer

    \(\frac{p(p−4)}{2(p−9)}\)

    Example \(\PageIndex{49}\)

    \(\frac{3q^2}{q^2+q−6}·\frac{q^2−9}{9q}\)

    Example \(\PageIndex{50}\)

    \(\frac{4r}{r^2−3r−10}·\frac{r^2−25}{8r^2}\)

    Answer

    \(\frac{r+5}{2r(r+2)}\)

    Example \(\PageIndex{51}\)

    \(\frac{s}{s^2−9s+14}·\frac{s^2−49}{7s^2}\)

    Example \(\PageIndex{52}\)

    \(\frac{x^2−7x}{x^2+6x+9}·\frac{x+3}{4x}\)

    Answer

    \(\frac{x−7}{4(x+3)}\)

    Example \(\PageIndex{53}\)

    \(\frac{2y^2−10y}{y^2+10y+25}·\frac{y+5}{6y}\)

    Example \(\PageIndex{54}\)

    \(\frac{z^2+3z}{z^2−3z−4}·\frac{z−4}{z^2}\)

    Answer

    \(\frac{z+3}{z(z+1)}\)

    Example \(\PageIndex{55}\)

    \(\frac{2a^2+8a}{a^2−9a+20}·\frac{a−5}{a^2}\)

    Example \(\PageIndex{56}\)

    \(\frac{28−4b}{3b−3}·\frac{b^2+8b−9}{b^2−49}\)

    Answer

    \(−\frac{4(b+9)}{3(b+7)}\)

    Example \(\PageIndex{57}\)

    \(\frac{18c−2c^2}{6c+30}·\frac{c^2+7c+10}{c^2−81}\)

    Example \(\PageIndex{58}\)

    \(\frac{35d−7d^2}{d^2+7d}·\frac{d^2+12d+35}{d^2−25}\)

    Answer

    −7

    Example \(\PageIndex{59}\)

    \(\frac{72m−12m^2}{8m+32}·\frac{m^2+10m+24}{m^2−36}\)

    Example \(\PageIndex{60}\)

    \(\frac{4n+20}{n^2+n−20}·\frac{n^2−16}{4n+16}\)

    Answer

    1

    Example \(\PageIndex{61}\)

    \(\frac{6p^2−6p}{p^2+7p−18}·\frac{p^2−81}{3p^2−27p}\)

    Example \(\PageIndex{62}\)

    \(\frac{q^2−2q}{q^2+6q−16}·\frac{q^2−64}{q^2−8q}\)

    Answer

    1

    Example \(\PageIndex{63}\)

    \(\frac{2r^2−2r}{r^2+4r−5}·\frac{r^2−25}{2r^2−10r}\)

    Divide Rational Expressions

    In the following exercises, divide.

    Example \(\PageIndex{64}\)

    \(\frac{t−6}{3−t}÷\frac{t^2−9}{t−5}\)

    Answer

    \(−\frac{2t}{t^3−5t−9}\)

    Example \(\PageIndex{65}\)

    \(\frac{v−5}{11−v}÷\frac{v^2−25}{v−11}\)

    Example \(\PageIndex{66}\)

    \(\frac{10+w}{w−8}÷\frac{100−w^2}{8−w}\)

    Answer

    \(−\frac{1}{10−w}\)

    Example \(\PageIndex{67}\)

    \(\frac{7+x}{x−6}÷\frac{49−x^2}{x+6}\)

    Example \(\PageIndex{68}\)

    \(\frac{27y^2}{3y−21}÷\frac{3y^2+18}{y^2+13y+42}\)

    Answer

    \(\frac{3y^2(y+6)(y+7)}{(y−7)(y2+6)}\)

    Example \(\PageIndex{69}\)

    \(\frac{24z^2}{2z−8}÷\frac{4z−28}{z^2−11z+28}\)​​​​​​​

    Example \(\PageIndex{70}\)

    \(\frac{16a^2}{4a+36}÷\frac{4a^2−24a}{a^2+4a−45}\)

    Answer

    \(\frac{a(a−5)}{a−6}\)

    Example \(\PageIndex{71}\)

    \(\frac{24b^2}{2b−4}÷\frac{12b^2+36b}{b^2−11b+18}\)

    Example \(\PageIndex{72}\)

    \(\frac{3c^2-16c+5}{c^2-25}÷\frac{3c^2-14c-5}{c^2+10c+25}\)

    Answer

    \(\frac{(3c-1)(c+5)}{(3c+1)(c−5)}\)​​​​​​​

    Example \(\PageIndex{73}\)

    \(\frac{2d^2+d−3}{d^2−16}÷\frac{2d^2−9d−18}{d^2−8d+16}\)

    Example \(\PageIndex{74}\)

    \(\frac{6m^2−13m+2}{9−m^2}÷\frac{6m^2+23m−4}{m^2−6m+9}\)

    Answer

    \(−\frac{(m−2)(m−3)}{(3+m)(m+4)}\)

    Example \(\PageIndex{75}\)

    \(\frac{2n^2−3n−14}{25−n^2}÷\frac{2n^2−13n+21}{n^2−10n+25}\)

    Example \(\PageIndex{76}\)

    \(\frac{3s^2}{s^2−16}÷\frac{s^3+4s^2+16s}{s^3−64}\)

    Answer

    \(\frac{3s}{s+4}\)

    Example \(\PageIndex{77}\)

    \(\frac{r^2−9}{15}÷\frac{r^3−27}{5r^2+15r+45}\)

    Example \(\PageIndex{78}\)

    \(\frac{p^3+q^3}{3p^2+3pq+3q^2}÷\frac{p^2−q^2}{12}\)

    Answer

    \(\frac{4(p^2−pq+q^2)}{(p−q)(p^2+pq+q^2)}\)

    Example \(\PageIndex{79}\)

    \(\frac{v^3−8w^3}{2v^2+4vw+8w^2}÷\frac{v^2−4w^2}{4}\)

    Example \(\PageIndex{80}\)

    \(\frac{t^2−9}{2t}÷(t^2−6t+9)\)

    Answer

    \(\frac{t+3}{2t(t−3)}\)

    Example \(\PageIndex{81}\)

    \(\frac{x^2+3x−10}{4x}÷(2x^2+20x+50)\)

    Example \(\PageIndex{82}\)

    \(\frac{2y^2−10yz−48z^2}{2y−1}÷(4y^2−32yz)\)

    Answer

    \(\frac{y+3z}{2y(2y−1)}\)

    Example \(\PageIndex{83}\)

    \(\frac{2m^2−98n^2}{2m+6}÷(m^2−7mn)\)

    Example \(\PageIndex{84}\)

    \(\frac{\frac{2a^2−a−21}{5a+20}}{\frac{a^2+7a+12}{a^2+8a+16}}\)

    Answer

    \(\frac{2a−7}{5}\)

    Example \(\PageIndex{85}\)

    \(\frac{\frac{3b^2+2b−8}{12b+18}}{\frac{3b^2+2b−8}{2b^2−7b−15}}\)

    Example \(\PageIndex{86}\)

    \(\frac{\frac{12c^2−12}{2c^2−3c+14}}{\frac{c+4}{6c^2−13c+5}}\)

    Answer

    3(3c−5)

    Example \(\PageIndex{87}\)

    \(\frac{\frac{4d^2+7d−2}{35d+10}}{\frac{d^2−4}{7d^2−12d−4}}\)​​​​​​​​​​​​​​

    Example \(\PageIndex{88}\)

    \(\frac{10m^2+80m}{3m−9}·\frac{m^2+4m−21}{m^2−9m+20}÷\frac{5m^2+10m}{2m−10}\)

    Answer

    \(\frac{4(m+8)(m+7)}{3(m−4)(m+2)}\)

    Example \(\PageIndex{89}\)

    \(\frac{4n^2+32n}{3n+2}·\frac{3n^2−n−2}{n^2+n−30}÷\frac{108n^2−24n}{n+6}\)

    Example \(\PageIndex{90}\)

    \(\frac{12p^2+3p}{p+3}÷\frac{p^2+2p−63}{p^2−p−12}·\frac{p−7}{9p^3−9p^2}\)

    Answer

    \(\frac{(4p+1)(p−7)}{3p(p+9)(p−1)}\)

    Example \(\PageIndex{91}\)

    \(\frac{6q+3}{9q^2−9q}÷\frac{q^2+14q+33}{q^2+4q−5}·\frac{4q^2+12q}{12q+6}\)​​​​​​​

    Everyday Math

    Example \(\PageIndex{92}\)

    Probability The director of large company is interviewing applicants for two identical jobs. If w= the number of women applicants and m= the number of men applicants, then the probability that two women are selected for the jobs is \(\frac{w}{w+m}·\frac{w−1}{w+m−1}\).

    1. Simplify the probability by multiplying the two rational expressions.
    2. Find the probability that two women are selected when w=5 and m=10.
    Answer
    1. \(\frac{w(w−1)}{(w+m)(w+m−1)}\)
    2. \(\frac{2}{21}\)
    Example \(\PageIndex{93}\)

    Area of a triangle The area of a triangle with base b and height h is \(\frac{bh}{2}\). If the triangle is stretched to make a new triangle with base and height three times as much as in the original triangle, the area is \(\frac{9bh}{2}\). Calculate how the area of the new triangle compares to the area of the original triangle by dividing \(\frac{9bh}{2}\) by \(\frac{bh}{2}\).​​​​​​​

    Writing Exercises

    Example \(\PageIndex{94}\)
    1. Multiply \(\frac{7}{4}·\frac{9}{10}\) and explain all your steps.
    2. Multiply \(\frac{n}{n−3}·\frac{9n+3}{n}\) and explain all your steps.
    3. Evaluate your answer to part (b) when n=7. Did you get the same answer you got in part (a)? Why or why not?
    Answer

    Answers will vary.​​​​​​​

    Example \(\PageIndex{95}\)
    1. Divide \(\frac{24}{5}÷6\) and explain all your steps.
    2. Divide \(\frac{x^2−1}{x}÷(x+1)\) and explain all your steps.
    3. Evaluate your answer to part (b) when x=5. Did you get the same answer you got in part (a)? Why or why not?​​​​​​​

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    The above image is a table with four columns and four 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 multiply rational expressions.”, the next row reads “divide rational expressions.”, the last row reads “after reviewing this checklist, what will you do to become confident for all objectives?” The remaining columns are blank.

    ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


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