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# 8.6E: Ejercicios

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### La práctica hace 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

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. $$2 a b \sqrt[3]{2 a}$$

##### Ejercicio SET B: Racionalizar un denominador de un término

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

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.$$\dfrac{2 \sqrt[4]{3 a^{2}}}{a}$$

##### Ejercicio SET C: Racionalizar un Denominador de Dos Plazos

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

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. $$\dfrac{(\sqrt{x}+2 \sqrt{2})^{2}}{x-8}$$

##### Ejercicio SET D: ejercicios de escritura
1. Simplifica$$\sqrt{\dfrac{27}{3}}$$ y explica todos tus pasos.
2. Simplifica$$\sqrt{\dfrac{27}{5}}$$ y explica todos tus pasos.
3. ¿Por qué son diferentes los dos métodos de simplificación de las raíces cuadradas?
1. Explique lo que se entiende por la palabra racionalizar en la frase, “racionalizar un denominador”.
2. Explicar por qué multiplicar$$\sqrt{2x}-3$$ por su conjugado resulta en una epxressión sin radicales.
3. Explique por qué multiplicar$$\dfrac{7}{\sqrt[3]{x}}$$ por$$\dfrac{\sqrt[3]{x}}{\sqrt[3]{x}}$$ no racionaliza el denominador.
Contestar

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?

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