8.6E: Ejercicios
- Page ID
- 51748
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)La práctica hace a la perfección
En los siguientes ejercicios, simplifique.
1. a. \(\dfrac{\sqrt{128}}{\sqrt{72}}\quad\) b. \(\dfrac{\sqrt[3]{128}}{\sqrt[3]{54}}\)
2. a. \(\dfrac{\sqrt{48}}{\sqrt{75}}\quad\) b. \(\dfrac{\sqrt[3]{81}}{\sqrt[3]{24}}\)
3. a.\(\dfrac{\sqrt{200 m^{5}}}{\sqrt{98 m}}\quad\) b. \(\dfrac{\sqrt[3]{54 y^{2}}}{\sqrt[3]{2 y^{5}}}\)
4. a. \(\dfrac{\sqrt{108 n^{7}}}{\sqrt{243 n^{3}}}\quad\) b. \(\dfrac{\sqrt[3]{54 y}}{\sqrt[3]{16 y^{4}}}\)
5. a. \(\dfrac{\sqrt{75 r^{3}}}{\sqrt{108 r^{7}}}\quad\) b. \(\dfrac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{81 x^{4}}}\)
6. a. \(\dfrac{\sqrt{196 q}}{\sqrt{484 q^{5}}}\quad\) b. \(\dfrac{\sqrt[3]{16 m^{4}}}{\sqrt[3]{54 m}}\)
7. a. \(\dfrac{\sqrt{108 p^{5} q^{2}}}{\sqrt{3 p^{3} q^{6}}}\quad\) b. \(\dfrac{\sqrt[3]{-16 a^{4} b^{-2}}}{\sqrt[3]{2 a^{-2} b}}\)
8. a. \(\dfrac{\sqrt{98 r s^{10}}}{\sqrt{2 r^{3} s^{4}}}\quad\) b. \(\dfrac{\sqrt[3]{-375 y^{4} z^{2}}}{\sqrt[3]{3 y^{-2} z^{4}}}\)
9. a. \(\dfrac{\sqrt{320 m n^{-5}}}{\sqrt{45 m^{-7} n^{3}}}\quad\) b. \(\dfrac{\sqrt[3]{16 x^{4} y^{-2}}}{\sqrt[3]{-54 x^{-2} y^{4}}}\)
10. a. \(\dfrac{\sqrt{810 c^{-3} d^{7}}}{\sqrt{1000 c d}}\quad\) b. \(\dfrac{\sqrt[3]{24 a^{7} b^{-1}}}{\sqrt[3]{-81 a^{-2} b^{2}}}\)
11. \(\dfrac{\sqrt{56 x^{5} y^{4}}}{\sqrt{2 x y^{3}}}\)
12. \(\dfrac{\sqrt{72 a^{3} b^{6}}}{\sqrt{3 a b^{3}}}\)
13. \(\dfrac{\sqrt[3]{48 a^{3} b^{6}}}{\sqrt[3]{3 a^{-1} b^{3}}}\)
14. \(\dfrac{\sqrt[3]{162 x^{-3} y^{6}}}{\sqrt[3]{2 x^{3} y^{-2}}}\)
- Contestar
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1. a. \(\dfrac{4}{3}\) b. \(\dfrac{4}{3}\)
3. a. \(\dfrac{10 m^{2}}{7}\) b. \(\dfrac{3}{y}\)
5. a. \(\dfrac{5}{6 r^{2}}\) b. \(\dfrac{2x}{3}\)
7. a. \(\dfrac{6 p}{q^{2}}\) b. \(-\dfrac{2 a^{2}}{b}\)
9. a. \(\dfrac{8 m^{4}}{3 n^{4}}\) b. \(-\dfrac{2 x^{2}}{3 y^{2}}\)
11. \(4 x^{4} \sqrt{7 y}\)
13. \(2 a b \sqrt[3]{2 a}\)
En los siguientes ejercicios, racionalizar el denominador.
15. a. \(\dfrac{10}{\sqrt{6}}\quad\) b. \(\sqrt{\dfrac{4}{27}}\quad\) c. \(\dfrac{10}{\sqrt{5 x}}\)
16. a. \(\dfrac{8}{\sqrt{3}}\quad\) b. \(\sqrt{\dfrac{7}{40}}\quad\) c. \(\dfrac{8}{\sqrt{2 y}}\)
17. a. \(\dfrac{6}{\sqrt{7}}\quad\) b. \(\sqrt{\dfrac{8}{45}}\quad\) c. \(\dfrac{12}{\sqrt{3 p}}\)
18. a. \(\dfrac{4}{\sqrt{5}}\quad\) b. \(\sqrt{\dfrac{27}{80}}\quad\) c. \(\dfrac{18}{\sqrt{6 q}}\)
19. a. \(\dfrac{1}{\sqrt[3]{5}}\quad\) b. \(\sqrt[3]{\dfrac{5}{24}}\quad\) c. \(\dfrac{4}{\sqrt[3]{36 a}}\)
20. a. \(\dfrac{1}{\sqrt[3]{3}}\quad\) b. \(\sqrt[3]{\dfrac{5}{32}}\quad\) c. \(\dfrac{7}{\sqrt[3]{49 b}}\)
21. a. \(\dfrac{1}{\sqrt[3]{11}}\quad\) b. \(\sqrt[3]{\dfrac{7}{54}}\quad\) c. \(\dfrac{3}{\sqrt[3]{3 x^{2}}}\)
22. a. \(\dfrac{1}{\sqrt[3]{13}}\quad\) b. \(\sqrt[3]{\dfrac{3}{128}}\quad\) c. \(\dfrac{3}{\sqrt[3]{6 y^{2}}}\)
23. a. \(\dfrac{1}{\sqrt[4]{7}}\quad\) b. \(\sqrt[4]{\dfrac{5}{32}}\quad\) c. \(\dfrac{4}{\sqrt[4]{4 x^{2}}}\)
24. a. \(\dfrac{1}{\sqrt[4]{4}}\quad\) b. \(\sqrt[4]{\dfrac{9}{32}}\quad\) c. \(\dfrac{6}{\sqrt[4]{9 x^{3}}}\)
25. a. \(\dfrac{1}{\sqrt[4]{9}}\quad\) b. \(\sqrt[4]{\dfrac{25}{128}}\quad\) c. \(\dfrac{6}{\sqrt[4]{27 a}}\)
26. a. \(\dfrac{1}{\sqrt[4]{8}}\quad\) b. \(\sqrt[4]{\dfrac{27}{128}}\quad\) c. \(\dfrac{16}{\sqrt[4]{64 b^{2}}}\)
- Contestar
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15. a. \(\dfrac{5 \sqrt{6}}{3}\) b. \(\dfrac{2 \sqrt{3}}{9}\) c. \(\dfrac{2 \sqrt{5 x}}{x}\)
17. a. \(\dfrac{6 \sqrt{7}}{7}\) b. \(\dfrac{2 \sqrt{10}}{15}\) c. \(\dfrac{4 \sqrt{3 p}}{p}\)
19. a. \(\dfrac{\sqrt[3]{25}}{5}\) b. \(\dfrac{\sqrt[3]{45}}{6}\) c. \(\dfrac{2 \sqrt[3]{6 a^{2}}}{3 a}\)
21. a. \(\dfrac{\sqrt[3]{121}}{11}\) b. \(\dfrac{\sqrt[3]{28}}{6}\) c. \(\dfrac{\sqrt[3]{9 x}}{x}\)
23. a. \(\dfrac{\sqrt[4]{343}}{7}\) b. \(\dfrac{\sqrt[4]{40}}{4}\) c. \(\dfrac{2 \sqrt[4]{4 x^{2}}}{x}\)
25. a. \(\dfrac{\sqrt[4]{9}}{3}\) b. \(\dfrac{\sqrt[4]{50}}{4}\) c. \(\dfrac{2 \sqrt[4]{3 a^{2}}}{a}\)
En los siguientes ejercicios, simplifique.
27. \(\dfrac{8}{1-\sqrt{5}}\)
28. \(\dfrac{7}{2-\sqrt{6}}\)
29. \(\dfrac{6}{3-\sqrt{7}}\)
30. \(\dfrac{5}{4-\sqrt{11}}\)
31. \(\dfrac{\sqrt{3}}{\sqrt{m}-\sqrt{5}}\)
32. \(\dfrac{\sqrt{5}}{\sqrt{n}-\sqrt{7}}\)
33. \(\dfrac{\sqrt{2}}{\sqrt{x}-\sqrt{6}}\)
34. \(\dfrac{\sqrt{7}}{\sqrt{y}+\sqrt{3}}\)
35. \(\dfrac{\sqrt{r}+\sqrt{5}}{\sqrt{r}-\sqrt{5}}\)
36. \(\dfrac{\sqrt{s}-\sqrt{6}}{\sqrt{s}+\sqrt{6}}\)
37. \(\dfrac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}\)
38. \(\dfrac{\sqrt{m}-\sqrt{3}}{\sqrt{m}+\sqrt{3}}\)
- Contestar
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27. \(-2(1+\sqrt{5})\)
29. \(3(3+\sqrt{7})\)
31. \(\dfrac{\sqrt{3}(\sqrt{m}+\sqrt{5})}{m-5}\)
33. \(\dfrac{\sqrt{2}(\sqrt{x}+\sqrt{6})}{x-6}\)
35. \(\dfrac{(\sqrt{r}+\sqrt{5})^{2}}{r-5}\)
37. \(\dfrac{(\sqrt{x}+2 \sqrt{2})^{2}}{x-8}\)
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- Simplifica \(\sqrt{\dfrac{27}{3}}\) y explica todos tus pasos.
- Simplifica \(\sqrt{\dfrac{27}{5}}\) y explica todos tus pasos.
- ¿Por qué son diferentes los dos métodos de simplificación de las raíces cuadradas?
- Explica lo que se entiende por la palabra racionalizar en la frase, “racionalizar un denominador”.
- Explicar por qué multiplicar \(\sqrt{2x}-3\) por su conjugado resulta en una epxressión sin radicales.
- Explicar por qué multiplicar \(\dfrac{7}{\sqrt[3]{x}}\) por \(\dfrac{\sqrt[3]{x}}{\sqrt[3]{x}}\) no racionaliza el denominador.
- Contestar
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1. Las respuestas variarán
3. Las respuestas variarán
Autocomprobación
a. Después de completar los ejercicios, utilice esta lista de verificación para evaluar su dominio de los objetivos de esta sección.
b. Después de mirar la lista de verificación, ¿cree que está bien preparado para la siguiente sección? ¿Por qué o por qué no?