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Mathematics LibreTexts

7.4E: Exercises

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    Simplify a Complex Rational Expression by Writing it as Division

    In the following exercises, simplify each complex rational expression by writing it as division.

    1. \(\dfrac{\dfrac{2 a}{a+4}}{\dfrac{4 a^{2}}{a^{2}-16}}\)

    Answer

    \(\dfrac{a-4}{2 a}\)

    2. \(\dfrac{\dfrac{3 b}{b-5}}{\dfrac{b^{2}}{b^{2}-25}}\)

    3. \(\dfrac{\dfrac{5}{c^{2}+5 c-14}}{\dfrac{10}{c+7}}\)

    Answer

    \(\dfrac{1}{2(c-2)}\)

    4. \(\dfrac{\dfrac{8}{d^{2}+9 d+18}}{\dfrac{12}{d+6}}\)

    5. \(\dfrac{\dfrac{1}{2}+\dfrac{5}{6}}{\dfrac{2}{3}+\dfrac{7}{9}}\)

    Answer

    \(\dfrac{12}{13}\)

    6. \(\dfrac{\dfrac{1}{2}+\dfrac{3}{4}}{\dfrac{3}{5}+\dfrac{7}{10}}\)

    7. \(\dfrac{\dfrac{2}{3}-\dfrac{1}{9}}{\dfrac{3}{4}+\dfrac{5}{6}}\)

    Answer

    \(\dfrac{20}{57}\)

    8. \(\dfrac{\dfrac{1}{2}-\dfrac{1}{6}}{\dfrac{2}{3}+\dfrac{3}{4}}\)

    9. \(\dfrac{\dfrac{n}{m}+\dfrac{1}{n}}{\dfrac{1}{n}-\dfrac{n}{m}}\)

    Answer

    \(\dfrac{n^{2}+m}{m-n^{2}}\)

    10. \(\dfrac{\dfrac{1}{p}+\dfrac{p}{q}}{\dfrac{q}{p}-\dfrac{1}{q}}\)

    11. \(\dfrac{\dfrac{1}{r}+\dfrac{1}{t}}{\dfrac{1}{r^{2}}-\dfrac{1}{t^{2}}}\)

    Answer

    \(\dfrac{r t}{t-r}\)

    12. \(\dfrac{\dfrac{2}{v}+\dfrac{2}{w}}{\dfrac{1}{v^{2}}-\dfrac{1}{w^{2}}}\)

    13. \(\dfrac{x-\dfrac{2 x}{x+3}}{\dfrac{1}{x+3}+\dfrac{1}{x-3}}\)

    Answer

    \(\dfrac{(x+1)(x-3)}{2}\)

    14. \(\dfrac{y-\dfrac{2 y}{y-4}}{\dfrac{2}{y-4}+\dfrac{2}{y+4}}\)

    15. \(\dfrac{2-\dfrac{2}{a+3}}{\dfrac{1}{a+3}+\dfrac{a}{2}}\)

    Answer

    \(\dfrac{4}{a+1}\)

    16. \(\dfrac{4+\dfrac{4}{b-5}}{\dfrac{1}{b-5}+\dfrac{b}{4}}\)

    Simplify a Complex Rational Expression by Using the LCD

    In the following exercises, simplify each complex rational expression by using the LCD.

    17. \(\dfrac{\dfrac{1}{3}+\dfrac{1}{8}}{\dfrac{1}{4}+\dfrac{1}{12}}\)

    Answer

    \(\dfrac{11}{8}\)

    18. \(\dfrac{\dfrac{1}{4}+\dfrac{1}{9}}{\dfrac{1}{6}+\dfrac{1}{12}}\)

    19. \(\dfrac{\dfrac{5}{6}+\dfrac{2}{9}}{\dfrac{7}{18}-\dfrac{1}{3}}\)

    Answer

    \(19\)

    20. \(\dfrac{\dfrac{1}{6}+\dfrac{4}{15}}{\dfrac{3}{5}-\dfrac{1}{2}}\)

    21. \(\dfrac{\dfrac{c}{d}+\dfrac{1}{d}}{\dfrac{1}{d}-\dfrac{d}{c}}\)

    Answer

    \(\dfrac{c^{2}+c}{c-d^{2}}\)

    22. \(\dfrac{\dfrac{1}{m}+\dfrac{m}{n}}{\dfrac{n}{m}-\dfrac{1}{n}}\)

    23. \(\dfrac{\dfrac{1}{p}+\dfrac{1}{q}}{\dfrac{1}{p^{2}}-\dfrac{1}{q^{2}}}\)

    Answer

    \(\dfrac{p q}{q-p}\)

    24. \(\dfrac{\dfrac{2}{r}+\dfrac{2}{t}}{\dfrac{1}{r^{2}}-\dfrac{1}{t^{2}}}\)

    25. \(\dfrac{\dfrac{2}{x+5}}{\dfrac{3}{x-5}+\dfrac{1}{x^{2}-25}}\)

    Answer

    \(\dfrac{2 x-10}{3 x+16}\)

    26. \(\dfrac{\dfrac{5}{y-4}}{\dfrac{3}{y+4}+\dfrac{2}{y^{2}-16}}\)

    27. \(\dfrac{\dfrac{5}{z^{2}-64}+\dfrac{3}{z+8}}{\dfrac{1}{z+8}+\dfrac{2}{z-8}}\)

    Answer

    \(\dfrac{3 z-19}{3 z+8}\)

    28. \(\dfrac{\dfrac{3}{s+6}+\dfrac{5}{s-6}}{\dfrac{1}{s^{2}-36}+\dfrac{4}{s+6}}\)

    29. \(\dfrac{\dfrac{4}{a^{2}-2 a-15}}{\dfrac{1}{a-5}+\dfrac{2}{a+3}}\)

    Answer

    \(\dfrac{4}{3 a-7}\)

    30. \(\dfrac{\dfrac{5}{b^{2}-6 b-27}}{\dfrac{3}{b-9}+\dfrac{1}{b+3}}\)

    31. \(\dfrac{\dfrac{5}{c+2}-\dfrac{3}{c+7}}{\dfrac{5 c}{c^{2}+9 c+14}}\)

    Answer

    \(\dfrac{2 c+29}{5 c}\)

    32. \(\dfrac{\dfrac{6}{d-4}-\dfrac{2}{d+7}}{\dfrac{2 d}{d^{2}+3 d-28}}\)

    33. \(\dfrac{2+\dfrac{1}{p-3}}{\dfrac{5}{p-3}}\)

    Answer

    \(\dfrac{2 p-5}{5}\)

    34. \(\dfrac{\dfrac{n}{n-2}}{3+\dfrac{5}{n-2}}\)

    35. \(\dfrac{\dfrac{m}{m+5}}{4+\dfrac{1}{m-5}}\)

    Answer

    \(\dfrac{m(m-5)}{(4 m-19)(m+5)}\)

    36. \(\dfrac{7+\dfrac{2}{q-2}}{\dfrac{1}{q+2}}\)

    In the following exercises, simplify each complex rational expression using either method.

    37. \(\dfrac{\dfrac{3}{4}-\dfrac{2}{7}}{\dfrac{1}{2}+\dfrac{5}{14}}\)

    Answer

    \(\dfrac{13}{24}\)

    38. \(\dfrac{\dfrac{v}{w}+\dfrac{1}{v}}{\dfrac{1}{v}-\dfrac{v}{w}}\)

    39. \(\dfrac{\dfrac{2}{a+4}}{\dfrac{1}{a^{2}-16}}\)

    Answer

    \(2(a-4)\)

    40. \(\dfrac{\dfrac{3}{b^{2}-3 b-40}}{\dfrac{5}{b+5}-\dfrac{2}{b-8}}\)

    41. \(\dfrac{\dfrac{3}{m}+\dfrac{3}{n}}{\dfrac{1}{m^{2}}-\dfrac{1}{n^{2}}}\)

    Answer

    \(\dfrac{3 m n}{n-m}\)

    42. \(\dfrac{\dfrac{2}{r-9}}{\dfrac{1}{r+9}+\dfrac{3}{r^{2}-81}}\)

    43. \(\dfrac{x-\dfrac{3 x}{x+2}}{\dfrac{3}{x+2}+\dfrac{3}{x-2}}\)

    Answer

    \(\dfrac{(x-1)(x-2)}{6}\)

    44. \(\dfrac{\dfrac{y}{y+3}}{2+\dfrac{1}{y-3}}\)

    Writing Exercises

    45. In this section, you learned to simplify the complex fraction \(\dfrac{\dfrac{3}{x+2}}{\dfrac{x}{x^{2}-4}}\) two ways: rewriting it as a division problem or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?

    Answer

    Answers will vary.

    44. Efraim wants to start simplifying the complex fraction \(\dfrac{\dfrac{1}{a}+\dfrac{1}{b}}{\dfrac{1}{a}-\dfrac{1}{b}}\) by cancelling the variables from the numerator and denominator, \(\dfrac{\dfrac{1}{\cancel{a}}+\dfrac{1}{\cancel {b}}}{\dfrac{1}{\cancel{a}}-\dfrac{1}{\cancel{b}}}\). Explain what is wrong with Efraim’s plan.

    7.4E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.