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Mathematics LibreTexts

7.2E: Exercises

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    30874
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    Practice Makes Perfect

    Determine the Values for Which a Rational Expression is Undefined

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

    1. a. \(\dfrac{2x^2}{z}\) b. \(\dfrac{4p−1}{6p−5}\) c. \(\dfrac{n−3}{n^2+2n−8}\)

    Answer

    a. \(z=0\)
    b. \(p=\dfrac{5}{6}\)
    c. \(n=−4, n=2\)

    2. a. \(\dfrac{10m}{11n}\) b. \(\dfrac{6y+13}{4y−9}\) c. \(\dfrac{b−8}{b^2−36}\)

    3. a. \(\dfrac{4x^2y}{3y}\) b. \(\dfrac{3x−2}{2x+1}\) c. \(\dfrac{u−1}{u^2−3u−28}\)

    Answer

    a. \(y=0\)
    b. \(x=−\dfrac{1}{2}\)
    c. \(u=−4, u=7\)

    4. a. \(\dfrac{5pq^2}{9q}\) b. \(\dfrac{7a−4}{3a+5}\) c. \(\dfrac{1}{x^2−4}\)

    Simplify Rational Expressions

    In the following exercises, simplify each rational expression.

    5. \(−\dfrac{44}{55}\)

    Answer

    \(−\dfrac{4}{5}\)

    6. \(\dfrac{56}{63}\)

    7. \(\dfrac{8m^3n}{12mn^2}\)

    Answer

    \(\dfrac{2m^2}{3n}\)

    8. \(\dfrac{36v^3w^2}{27vw^3}\)

    9. \(\dfrac{8n−96}{3n−36}\)

    Answer

    \(\dfrac{8}{3}\)

    10. \(\dfrac{12p−240}{5p−100}\)

    11. \(\dfrac{x^2+4x−5}{x^2−2x+1}\)

    Answer

    \(\dfrac{x+5}{x−1}\)

    12. \(\dfrac{y^2+3y−4}{y^2−6y+5}\)

    13. \(\dfrac{a^2−4}{a^2+6a−16}\)

    Answer

    \(\dfrac{a+2}{a+8}\)

    14. \(\dfrac{y^2−2y−3}{y^2−9}\)

    15. \(\dfrac{p^3+3p^2+4p+12}{p^2+p−6}\)

    Answer

    \(\dfrac{p^2+4}{p−2}\)

    16. \(\dfrac{x^3−2x^2−25x+50}{x^2−25}\)

    17. \(\dfrac{8b^2−32b}{2b^2−6b−80}\)

    Answer

    \(\dfrac{4b(b−4)}{(b+5)(b−8)}\)

    18. \(\dfrac{−5c^2−10c}{−10c^2+30c+100}\)

    19. \(\dfrac{3m^2+30mn+75n^2}{4m^2−100n^2}\)

    Answer

    \(\dfrac{3(m+5n)}{4(m−5n)}\)

    20. \(\dfrac{5r^2+30rs−35s^2}{r^2−49s^2}\)

    21. \(\dfrac{a−5}{5−a}\)

    Answer

    \(−1\)

    22. \(\dfrac{5−d}{d−5}\)

    23. \(\dfrac{20−5y}{y^2−16}\)

    Answer

    \(\dfrac{−5}{y+4}\)

    24. \(\dfrac{4v−32}{64−v^2}\)

    25. \(\dfrac{w^3+21}{6w^2−36}\)

    Answer

    \(\dfrac{w^2−6w+3}{6w−6}\)

    26. \(\dfrac{v^3+125}{v^2−25}\)

    27. \(\dfrac{z^2−9z+20}{16−z^2}\)

    Answer

    \(\dfrac{−z−5}{4+z}\)

    28. \(\dfrac{a^2−5z−36}{81−a^2}\)

    Multiply Rational Expressions

    In the following exercises, multiply the rational expressions.

    29. \(\dfrac{12}{16}·\dfrac{4}{10}\)

    Answer

    \(\dfrac{3}{10}\)

    30. \(\dfrac{3}{25}·\dfrac{16}{24}\)

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

    Answer

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

    32. \(\dfrac{12a^3b}{b^2}·\dfrac{2ab^2}{9b^3}\)

    33. \(\dfrac{5p^2}{p^2−5p−36}·\dfrac{p^2−16}{10p}\)

    Answer

    \(\dfrac{p(p−4)}{2(p−9)}\)

    34. \(\dfrac{3q^2}{q^2+q−6}·\dfrac{q^2−9}{9q}\)

    35. \(\dfrac{2y^2−10y}{y^2}+\dfrac{10y+2}{5}·\dfrac{y+5}{6y}\)

    Answer

    \(\dfrac{y−5}{3(y+5)}\)

    36. \(\dfrac{z^2+3z}{z^2−3z−4}·\dfrac{z−4}{z^2}\)

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

    Answer

    \(\dfrac{−4(b+9)}{3(b+7)}\)

    38. \(\dfrac{72m−12m^2}{8m+32}·\dfrac{m^2+10m+2}{4m^2−36}\)

    39. \(\dfrac{5c^2+9c+2}{c^2−25}·\dfrac{c^2+10c+25}{3c^2−14c−5}\)

    Answer

    \(\dfrac{(c+2)(c+2)}{(c−2)(c−3)}\)

    40. \(\dfrac{2d^2+d−3}{d^2−16}·\dfrac{d^2−8d+16}{2d^2−9d−18}\)

    41. \(\dfrac{2m^2−3m−2}{2m2+7m+3}·\dfrac{3m^2−14m+15}{3m^2+17m−20}\)

    Answer

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

    42. \(\dfrac{2n^2−3n−14}{25−n^2}·\dfrac{n^2−10n+25}{2n^2−13n+21}\)

    Divide Rational Expressions

    In the following exercises, divide the rational expressions.

    43. \(\dfrac{v−5}{11−v}÷\dfrac{v^2−25}{v−11}\)

    Answer

    \(−\dfrac{1}{v+5}\)

    44. \(\dfrac{10+w}{w−8}÷\dfrac{100−w^2}{8−w}\)

    45. \(\dfrac{3s^2}{s^2−16}÷\dfrac{s^3−4s^2+16s}{s^3−64}\)

    Answer

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

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

    47. \(\dfrac{p^3+q^3}{3p^2+3pq+3q^2}÷\dfrac{p^2−q^2}{12}\)

    Answer

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

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

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

    Answer

    \(\dfrac{x−2}{8x}\)

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

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

    Answer

    \(\dfrac{2a−7}{5}\)

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

    53. \(\dfrac{\dfrac{12c^2−12}{2c^2−3c+1}}{\dfrac{4c+4}{6c^2−13c+5}}\)

    Answer

    \(3(3c−5)\)

    54. \(\dfrac{\dfrac{4d^2+7d−2}{35d+10}}{\dfrac{d^2−4}{7d^2−12d−4}}\)

    For the following exercises, perform the indicated operations.

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

    Answer

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

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

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

    Answer

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

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

    Multiply and Divide Rational Functions

    In the following exercises, find the domain of each function.

    59. \(R(x)=\dfrac{x^3−2x^2−25x+50}{x^2−25}\)

    Answer

    \(x\neq 5\) and \(x\neq −5\)

    60. \(R(x)=\dfrac{x^3+3x^2−4x−12}{x^2−4}\)

    61. \(R(x)=\dfrac{3x^2+15x}{6x^2+6x−36}\)

    Answer

    \(x\neq 2\) and \(x\neq −3\)

    62. \(R(x)=\dfrac{8x^2−32x}{2x^2−6x−80}\)

    For the following exercises, find \(R(x)=f(x)·g(x)\) where \(f(x)\) and \(g(x)\) are given.

    63. \(f(x)=\dfrac{6x^2−12x}{x^2+7x−18} \quad g(x)=\dfrac{x^2−81}{3x^2−27x}\)

    Answer

    \(R(x)=2\)

    64. \(f(x)=\dfrac{x^2−2x}{x^2+6x−16} \quad g(x)=\dfrac{x^2−64}{x^2−8x}\)

    65. \(f(x)=\dfrac{4x}{x^2−3x−10} \quad g(x)=\dfrac{x^2−25}{8x^2}\)

    Answer

    \(R(x)=\dfrac{x+5}{2x(x+2)}\)

    66. \(f(x)=\dfrac{2x^2+8x}{x^2−9x+20} \quad g(x)=\dfrac{x−5}{x^2}\)

    For the following exercises, find \(R(x)=f(x)g(x)\) where \(f(x)\) and \(g(x)\) are given.

    67. \(f(x)=\dfrac{27x^2}{3x−21} \quad g(x)=\dfrac{3x^2+18x}{x^2+13x+42}\)

    Answer

    \(R(x)=\dfrac{3x(x+7)}{x−7}\)

    68. \(f(x)=\dfrac{24x^2}{2x−8} \quad g(x)=\dfrac{4x^3+28x^2}{x^2+11x+28}\)

    69. \(f(x)=\dfrac{16x^2}{4x+36} \quad g(x)=\dfrac{4x^2−24x}{x^2+4x−45}\)

    Answer

    \(R(x)=\dfrac{x(x−5)}{x−6}\)

    70. \(f(x)=\dfrac{24x^2}{2x−4} \quad g(x)=\dfrac{12x^2+36x}{x^2−11x+18}\)

    Writing Exercises

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

    Answer

    Answers will vary.

    72. Explain all the steps you take to simplify the rational expression \(\dfrac{p^2+4p−21}{9−p^2}\).

    73. 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.

    74. a. Divide \(\dfrac{24}{5}÷6\) and explain all your steps.
    b. Divide \(\dfrac{x^2−1}{x}÷(x+1)\) and explain all your steps.
    c. 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

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

    This table has four columns and six rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was determine the values for which a rational expression is undefined. In row 3, the I can was simplify rationale expressions. In row 4, the I can was multiply rational expressions. In row 5, the I can was divide rational expressions. In row 6, the I can was multiply and divide rational functions. There is the nothing in the other columns.

    b. If most of your checks were:

    …confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

    …with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

    …no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.