7.4E: Ejercicios
- Page ID
- 112736
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Simplifique 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 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. \(\dfrac{\dfrac{c}{d}+\dfrac{1}{d}}{\dfrac{1}{d}-\dfrac{d}{c}}\)
- Responder
-
\(\dfrac{c^{2}+c}{c-d^{2}}\)
22. \(\dfrac{\dfrac{1}{m}+\dfrac{m}{n}}{\dfrac{n}{m}-\dfrac{1}{n}}\)
23. \(\dfrac{\dfrac{1}{p}+\dfrac{1}{q}}{\dfrac{1}{p^{2}}-\dfrac{1}{q^{2}}}\)
- Responder
-
\(\dfrac{p q}{q-p}\)
24. \(\dfrac{\dfrac{2}{r}+\dfrac{2}{t}}{\dfrac{1}{r^{2}}-\dfrac{1}{t^{2}}}\)
25. \(\dfrac{\dfrac{2}{x+5}}{\dfrac{3}{x-5}+\dfrac{1}{x^{2}-25}}\)
- Responder
-
\(\dfrac{2 x-10}{3 x+16}\)
26. \(\dfrac{\dfrac{5}{y-4}}{\dfrac{3}{y+4}+\dfrac{2}{y^{2}-16}}\)
27. \(\dfrac{\dfrac{5}{z^{2}-64}+\dfrac{3}{z+8}}{\dfrac{1}{z+8}+\dfrac{2}{z-8}}\)
- Responder
-
\(\dfrac{3 z-19}{3 z+8}\)
28. \(\dfrac{\dfrac{3}{s+6}+\dfrac{5}{s-6}}{\dfrac{1}{s^{2}-36}+\dfrac{4}{s+6}}\)
29. \(\dfrac{\dfrac{4}{a^{2}-2 a-15}}{\dfrac{1}{a-5}+\dfrac{2}{a+3}}\)
- Responder
-
\(\dfrac{4}{3 a-7}\)
30. \(\dfrac{\dfrac{5}{b^{2}-6 b-27}}{\dfrac{3}{b-9}+\dfrac{1}{b+3}}\)
31. \(\dfrac{\dfrac{5}{c+2}-\dfrac{3}{c+7}}{\dfrac{5 c}{c^{2}+9 c+14}}\)
- Responder
-
\(\dfrac{2 c+29}{5 c}\)
32. \(\dfrac{\dfrac{6}{d-4}-\dfrac{2}{d+7}}{\dfrac{2 d}{d^{2}+3 d-28}}\)
33. \(\dfrac{2+\dfrac{1}{p-3}}{\dfrac{5}{p-3}}\)
- Responder
-
\(\dfrac{2 p-5}{5}\)
34. \(\dfrac{\dfrac{n}{n-2}}{3+\dfrac{5}{n-2}}\)
35. \(\dfrac{\dfrac{m}{m+5}}{4+\dfrac{1}{m-5}}\)
- Responder
-
\(\dfrac{m(m-5)}{(4 m-19)(m+5)}\)
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. \(\dfrac{\dfrac{3}{4}-\dfrac{2}{7}}{\dfrac{1}{2}+\dfrac{5}{14}}\)
- Responder
-
\(\dfrac{13}{24}\)
38. \(\dfrac{\dfrac{v}{w}+\dfrac{1}{v}}{\dfrac{1}{v}-\dfrac{v}{w}}\)
39. \(\dfrac{\dfrac{2}{a+4}}{\dfrac{1}{a^{2}-16}}\)
- Responder
-
\(2(a-4)\)
40. \(\dfrac{\dfrac{3}{b^{2}-3 b-40}}{\dfrac{5}{b+5}-\dfrac{2}{b-8}}\)
41. \(\dfrac{\dfrac{3}{m}+\dfrac{3}{n}}{\dfrac{1}{m^{2}}-\dfrac{1}{n^{2}}}\)
- Responder
-
\(\dfrac{3 m n}{n-m}\)
42. \(\dfrac{\dfrac{2}{r-9}}{\dfrac{1}{r+9}+\dfrac{3}{r^{2}-81}}\)
43. \(\dfrac{x-\dfrac{3 x}{x+2}}{\dfrac{3}{x+2}+\dfrac{3}{x-2}}\)
- Responder
-
\(\dfrac{(x-1)(x-2)}{6}\)
44. \(\dfrac{\dfrac{y}{y+3}}{2+\dfrac{1}{y-3}}\)
Ejercicios de escritura
45. En esta sección 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 denominador por la LCD. ¿Qué método prefieres? ¿Por qué?
- Responder
-
Las respuestas variarán.
44. Efraim quiere comenzar 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,\(\dfrac{\dfrac{1}{\cancel{a}}+\dfrac{1}{\cancel {b}}}{\dfrac{1}{\cancel{a}}-\dfrac{1}{\cancel{b}}}\). Explique qué tiene de malo el plan de Efraim.