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7.4E: Ejercicios

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    51810
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    Simplifica una expresión racional compleja escribiéndola como división

    En los siguientes ejercicios, simplifique cada expresión racional compleja escribiéndola como división.

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

    Simplifique una expresión racional compleja mediante el uso de la pantalla LCD

    En los siguientes ejercicios, simplifique cada expresión racional compleja mediante el uso de la pantalla LCD.

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

    Contestar

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

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

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

    Contestar

    \(19\)

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

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

    Contestar

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

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

    En los siguientes ejercicios, simplifique cada expresión racional compleja utilizando cualquiera de los dos métodos.

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

    Contestar

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

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

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

    Contestar

    \(2(a-4)\)

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

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

    Contestar

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

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

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

    Contestar

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

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

    Ejercicios de escritura

    45. En este apartado aprendiste a simplificar la fracción compleja de \(\dfrac{\dfrac{3}{x+2}}{\dfrac{x}{x^{2}-4}}\) dos maneras: reescribirla como problema de división o multiplicar el numerador y el denominador por el LCD. ¿Qué método prefieres? ¿Por qué?

    Contestar

    Las respuestas variarán.

    44. Efraim quiere empezar a simplificar la fracción compleja \(\dfrac{\dfrac{1}{a}+\dfrac{1}{b}}{\dfrac{1}{a}-\dfrac{1}{b}}\) cancelando las variables del numerador y denominador, \(\dfrac{\dfrac{1}{\cancel{a}}+\dfrac{1}{\cancel {b}}}{\dfrac{1}{\cancel{a}}-\dfrac{1}{\cancel{b}}}\). Explica qué es lo que está mal con el plan de Efraim.


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