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5.4E: Exercises

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    Practice Makes Perfect

    Undefined Values

    In the following exercises, determine the values for which the rational expression is undefined.

    1. \(\dfrac{2x^2}{z}\)
    2. \(\dfrac{4p−1}{6p−5}\)
    3. \(\dfrac{n−3}{n^2+2n−8}\)
    4. \(\dfrac{4x^2y}{3y}\)
    5. \(\dfrac{3x−2}{2x+1}\)
    6. \(\dfrac{u−1}{u^2−3u−28}\)
    Answer
    1. \(z=0\)
    2. \(p=\dfrac{5}{6}\)
    3. \(n=−4, n=2\)
    4. \(y=0\)
    5. \(x=−\dfrac{1}{2}\)
    6. \(u=−4, u=7\)

    Simplify Rational Expressions

    In the following exercises, simplify each rational expression.

    1. \(\dfrac{x+1+y}{y+x+1}\)
    2. \(\dfrac{8m^3n}{12mn^2}\)
    3. \(\dfrac{8n−96}{3n−36}\)
    4. \(\dfrac{x^2+4x−5}{x^2−2x+1}\)
    5. \(\dfrac{a^2−4}{a^2+6a−16}\)
    6. \(\dfrac{p^3+3p^2+4p+12}{p^2+p−6}\)
    7. \(\dfrac{8b^2−32b}{2b^2−6b−80}\)
    8. \(\dfrac{3m^2+30mn+75n^2}{4m^2−100n^2}\)
    9. \(\dfrac{a−5}{5−a}\)
    10. \(\dfrac{20−5y}{y^2−16}\)
    11. \(\dfrac{z^2−9z+20}{16−z^2}\)
    Answer
    1. 1
    2. \(\dfrac{2m^2}{3n}\)
    3. \(\dfrac{8}{3}\)
    4. \(\dfrac{x+5}{x−1}\)
    5. \(\dfrac{a+2}{a+8}\)
    6. \(\dfrac{p^2+4}{p−2}\)
    7. \(\dfrac{4b(b−4)}{(b+5)(b−8)}\)
    8. \(\dfrac{3(m+5n)}{4(m−5n)}\)
    9. \(−1\)
    10. \(\dfrac{−5}{y+4}\)
    11. \(-\dfrac{z-5}{4+z}\)

    Multiply Rational Expressions

    In the following exercises, multiply the rational expressions.

    Multiply Rational Expressions

    In the following exercises, multiply the rational expressions.

    1. \(\dfrac{12}{16}\cdot \dfrac{4}{10}\)
    2. \(\dfrac{5x^2y^4}{12xy^3}\cdot \dfrac{6x^2}{20y^2}\)
    3. \(\dfrac{5p^2}{p^2−5p−36}\cdot \dfrac{p^2−16}{10p}\)
    4. \(\dfrac{2y^2−10y}{y^2+10y+25}\cdot \dfrac{y+5}{6y}\)
    5. \(\dfrac{28−4b}{3b−3}\cdot \dfrac{b^2+8b−9}{b^2−49}\)
    6. \(\dfrac{c^2-10c+25}{c^2−25}\cdot \dfrac{c^2+10c+25}{3c^2−14c−5}\)
    7. \(\dfrac{2m^2−3m−2}{2m^2+7m+3}\cdot \dfrac{3m^2−14m+15}{3m^2+17m−20}\)
    Answer
    1. \(\dfrac{3}{10}\)
    2. \(\dfrac{x^3}{8y}\)
    3. \(\dfrac{p(p−4)}{2(p−9)}\)
    4. \(\dfrac{y−5}{3(y+5)}\)
    5. \(\dfrac{−4(b+9)}{3(b+7)}\)
    6. \(\dfrac{c+5}{3c+1}\)
    7. \(\dfrac{(m−3)(m−2)(3m-5)}{(m-1)(m+3)(3m+20)}\)
    Divide Rational Expressions

    In the following exercises, divide the rational expressions.

    1. \(\dfrac{v−5}{11−v}÷\dfrac{v^2−25}{v−11}\)
    2. \(\dfrac{3s^2}{s^2−16}÷\dfrac{s^3+4s^2+16s}{s^3−64}\)
    3. \(\dfrac{p^3+q^3}{3p^2+3pq+3q^2}÷\dfrac{p^2−q^2}{12}\)
    4. \(\dfrac{x^2+3x−10}{4x}÷(2x^2+20x+50)\)
    5. \(\dfrac{\dfrac{2a^2−a−21}{5a+20}}{\dfrac{a^2+7a+12}{a^2+8a+16}}\)
    6. \(\dfrac{\dfrac{12c^2−12}{2c^2−3c+1}}{\dfrac{4c+4}{6c^2−13c+5}}\)
    7. \(\dfrac{10m^2+80m}{3m−9}·\dfrac{m^2+4m−21}{m^2−9m+20}÷\dfrac{5m^2+10m}{2m−10}\)
    8. \(\dfrac{12p^2+3p}{p+3}÷\dfrac{p^2+2p−63}{p^2−p−12}·\dfrac{p−7}{9p^3−9p^2}\)
    Answer
    1. \(−\dfrac{1}{v+5}\)
    2. \(\dfrac{3s}{s+4}\)
    3. \(\dfrac{4(p^2−pq+q^2)}{(p−q)(p^2+pq+q^2)}\)
    4. \(\dfrac{x−2}{8x(x+5)}\)
    5. \(\dfrac{2a−7}{5}\)
    6. \(3(3c−5)\)
    7. \(\dfrac{4(m+8)(m+7)}{3(m−4)(m+2)}\)
    8. \(\dfrac{(4p+1)(p−4)}{3p(p+9)(p−1)}\)
    Writing Exercises

    33. Explain how you find the values of x for which the rational expression \(\dfrac{x^2−x−20}{x^2−4}\) is undefined.

    34.

    a. Multiply \(\dfrac{7}{4}·\dfrac{9}{10}\) and explain all your steps.

    b. Multiply \(\dfrac{n}{n−3}·\dfrac{9}{n+3}\) and explain all your steps.

    c. 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.

    This page titled 5.4E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Stanislav A. Trunov and Elizabeth J. Hale via source content that was edited to the style and standards of the LibreTexts platform.