Saltar al contenido principal

# 8.5E: Ejercicios

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

### La práctica hace a la perfección

##### Ejercicio A: sumar y restar expresiones radicales

En los siguientes ejercicios, simplifique. Supongamos que todas las variables son mayores o iguales a cero para que no se necesiten valores absolutos.

1. a. $$8 \sqrt{2}-5 \sqrt{2}\quad$$ b. $$5 \sqrt[3]{m}+2 \sqrt[3]{m}\quad$$ c. $$8 \sqrt[4]{m}-2 \sqrt[4]{n}$$

2. a. $$7 \sqrt{2}-3 \sqrt{2}\quad$$ b. $$7 \sqrt[3]{p}+2 \sqrt[3]{p}\quad$$ c. $$5 \sqrt[3]{x}-3 \sqrt[3]{x}$$

3. a. $$3 \sqrt{5}+6 \sqrt{5}\quad$$ b. $$9 \sqrt[3]{a}+3 \sqrt[3]{a}\quad$$ c. $$5 \sqrt[4]{2 z}+\sqrt[4]{2 z}$$

4. a. $$4 \sqrt{5}+8 \sqrt{5} \quad$$ b. $$\sqrt[3]{m}-4 \sqrt[3]{m} \quad$$ c. $$\sqrt{n}+3 \sqrt{n}$$

5. a. $$3 \sqrt{2 a}-4 \sqrt{2 a}+5 \sqrt{2 a} \quad$$ b. $$5 \sqrt[4]{3 a b}-3 \sqrt[4]{3 a b}-2 \sqrt[4]{3 a b}$$

6. a. $$\sqrt{11 b}-5 \sqrt{11 b}+3 \sqrt{11 b} \quad$$ b. $$8 \sqrt[4]{11 c d}+5 \sqrt[4]{11 c d}-9 \sqrt[4]{11 c d}$$

7. a. $$8 \sqrt{3 c}+2 \sqrt{3 c}-9 \sqrt{3 c} \quad$$ b. $$2 \sqrt[3]{4 p q}-5 \sqrt[3]{4 p q}+4 \sqrt[3]{4 p q}$$

8. a. $$3 \sqrt{5 d}+8 \sqrt{5 d}-11 \sqrt{5 d} \quad$$ b. $$11 \sqrt[3]{2 r s}-9 \sqrt[3]{2 r s}+3 \sqrt[3]{2 r s}$$

9. a. $$\sqrt{27}-\sqrt{75} \quad$$ b. $$\sqrt[3]{40}-\sqrt[3]{320} \quad$$ c. $$\frac{1}{2} \sqrt[4]{32}+\frac{2}{3} \sqrt[4]{162}$$

10. a. $$\sqrt{72}-\sqrt{98} \quad$$ b. $$\sqrt[3]{24}+\sqrt[3]{81} \quad$$ c. $$\frac{1}{2} \sqrt[4]{80}-\frac{2}{3} \sqrt[4]{405}$$

11. a. $$\sqrt{48}+\sqrt{27} \quad$$ b. $$\sqrt[3]{54}+\sqrt[3]{128} \quad$$ c. $$6 \sqrt[4]{5}-\frac{3}{2} \sqrt[4]{320}$$

12. a. $$\sqrt{45}+\sqrt{80} \quad$$ b. $$\sqrt[3]{81}-\sqrt[3]{192} \quad$$ c. $$\frac{5}{2} \sqrt[4]{80}+\frac{7}{3} \sqrt[4]{405}$$

13. a. $$\sqrt{72 a^{5}}-\sqrt{50 a^{5}} \quad$$ b. $$9 \sqrt[4]{80 p^{4}}-6 \sqrt[4]{405 p^{4}}$$

14. a. $$\sqrt{48 b^{5}}-\sqrt{75 b^{5}} \quad$$ b. $$8 \sqrt[3]{64 q^{6}}-3 \sqrt[3]{125 q^{6}}$$

15. a. $$\sqrt{80 c^{7}}-\sqrt{20 c^{7}} \quad$$ b. $$2 \sqrt[4]{162 r^{10}}+4 \sqrt[4]{32 r^{10}}$$

16. a. $$\sqrt{96 d^{9}}-\sqrt{24 d^{9}} \quad$$ b. $$5 \sqrt[4]{243 s^{6}}+2 \sqrt[4]{3 s^{6}}$$

17. $$3 \sqrt{128 y^{2}}+4 y \sqrt{162}-8 \sqrt{98 y^{2}}$$

18. $$3 \sqrt{75 y^{2}}+8 y \sqrt{48}-\sqrt{300 y^{2}}$$
Contestar

1. a. $$3 \sqrt{2}$$ b. $$7 \sqrt[3]{m}$$ c. $$6 \sqrt[4]{m}$$

3. a. $$9 \sqrt{5}$$ b. $$12 \sqrt[3]{a}$$ c. $$6 \sqrt[4]{2 z}$$

5. a. $$4 \sqrt{2 a}$$ b. $$0$$

7. a. $$\sqrt{3c}$$ b. $$\sqrt[3]{4 p q}$$

9. a. $$-2 \sqrt{3}$$ b. $$-2 \sqrt[3]{5}$$ c. $$3 \sqrt[4]{2}$$

11. a. $$7 \sqrt{3}$$ b. $$7 \sqrt[3]{2}$$ c. $$3 \sqrt[4]{5}$$

13. a. $$a^{2} \sqrt{2 a}$$ b. $$0$$

15. a. $$2 c^{3} \sqrt{5 c}$$ b. $$14 r^{2} \sqrt[4]{2 r^{2}}$$

17. $$4 y \sqrt{2}$$

##### Ejercicio B: multiplicar expresiones radicales

En los siguientes ejercicios, simplifique.

1. $$(-2 \sqrt{3})(3 \sqrt{18})$$

2. $$(8 \sqrt[3]{4})(-4 \sqrt[3]{18})$$

3.

1. $$(-4 \sqrt{5})(5 \sqrt{10})$$

2. $$(-2 \sqrt[3]{9})(7 \sqrt[3]{9})$$

3.

1. $$(5 \sqrt{6})(-\sqrt{12})$$

2. $$(-2 \sqrt[4]{18})(-\sqrt[4]{9})$$

3.

1. $$(-2 \sqrt{7})(-2 \sqrt{14})$$

2. $$(-3 \sqrt[4]{8})(-5 \sqrt[4]{6})$$

3.

1. $$\left(4 \sqrt{12 z^{3}}\right)(3 \sqrt{9 z})$$

2. $$\left(5 \sqrt[3]{3 x^{3}}\right)\left(3 \sqrt[3]{18 x^{3}}\right)$$

3.

1. $$\left(3 \sqrt{2 x^{3}}\right)\left(7 \sqrt{18 x^{2}}\right)$$

2. $$\left(-6 \sqrt[3]{20 a^{2}}\right)\left(-2 \sqrt[3]{16 a^{3}}\right)$$

3.

1. $$\left(-2 \sqrt{7 z^{3}}\right)\left(3 \sqrt{14 z^{8}}\right)$$

2. $$\left(2 \sqrt[4]{8 y^{2}}\right)\left(-2 \sqrt[4]{12 y^{3}}\right)$$

3.

1. $$\left(4 \sqrt{2 k^{5}}\right)\left(-3 \sqrt{32 k^{6}}\right)$$

2. $$\left(-\sqrt[4]{6 b^{3}}\right)\left(3 \sqrt[4]{8 b^{3}}\right)$$

3.
Contestar

1.

1. $$-18 \sqrt{6}$$

2. $$-64 \sqrt[3]{9}$$

3.

3.

1. $$-30 \sqrt{2}$$

2. $$6 \sqrt[4]{2}$$

3.

5.

1. $$72 z^{2} \sqrt{3}$$

2. $$45 x^{2} \sqrt[3]{2}$$

3.

7.

1. $$-42 z^{5} \sqrt{2 z}$$

2. $$-8 y \sqrt[4]{6 y}$$
##### Ejercicio C: utilizar multiplicación polinómica para multiplicar expresiones radicales

En los siguientes ejercicios, multiplica.

1. $$\sqrt{7}(5+2 \sqrt{7})$$

2. $$\sqrt[3]{6}(4+\sqrt[3]{18})$$

3.

1. $$\sqrt{11}(8+4 \sqrt{11})$$

2. $$\sqrt[3]{3}(\sqrt[3]{9}+\sqrt[3]{18})$$

3.

1. $$\sqrt{11}(-3+4 \sqrt{11})$$

2. $$\sqrt[4]{3}(\sqrt[4]{54}+\sqrt[4]{18})$$

3.

1. $$\sqrt{2}(-5+9 \sqrt{2})$$

2. $$\sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})$$

3.

1. $$(7+\sqrt{3})(9-\sqrt{3})$$

2. $$(8-\sqrt{2})(3+\sqrt{2})$$

1. $$(9-3 \sqrt{2})(6+4 \sqrt{2})$$

2. $$(\sqrt[3]{x}-3)(\sqrt[3]{x}+1)$$

3.

1. $$(3-2 \sqrt{7})(5-4 \sqrt{7})$$

2. $$(\sqrt[3]{x}-5)(\sqrt[3]{x}-3)$$

3.

1. $$(1+3 \sqrt{10})(5-2 \sqrt{10})$$

2. $$(2 \sqrt[3]{x}+6)(\sqrt[3]{x}+1)$$

3.

1. $$(7-2 \sqrt{5})(4+9 \sqrt{5})$$

2. $$(3 \sqrt[3]{x}+2)(\sqrt[3]{x}-2)$$

3.

3. $$(\sqrt{3}+\sqrt{10})(\sqrt{3}+2 \sqrt{10})$$

4. $$(\sqrt{11}+\sqrt{5})(\sqrt{11}+6 \sqrt{5})$$

5. $$(2 \sqrt{7}-5 \sqrt{11})(4 \sqrt{7}+9 \sqrt{11})$$

6. $$(4 \sqrt{6}+7 \sqrt{13})(8 \sqrt{6}-3 \sqrt{13})$$

1. $$(3+\sqrt{5})^{2}$$

2. $$(2-5 \sqrt{3})^{2}$$

3.

1. $$(4+\sqrt{11})^{2}$$

2. $$(3-2 \sqrt{5})^{2}$$

3.

1. $$(9-\sqrt{6})^{2}$$

2. $$(10+3 \sqrt{7})^{2}$$

3.

1. $$(5-\sqrt{10})^{2}$$

2. $$(8+3 \sqrt{2})^{2}$$

3.

7. $$(4+\sqrt{2})(4-\sqrt{2})$$

8. $$(7+\sqrt{10})(7-\sqrt{10})$$

9. $$(4+9 \sqrt{3})(4-9 \sqrt{3})$$

10. $$(1+8 \sqrt{2})(1-8 \sqrt{2})$$

11. $$(12-5 \sqrt{5})(12+5 \sqrt{5})$$

12. $$(9-4 \sqrt{3})(9+4 \sqrt{3})$$

13. $$(\sqrt[3]{3 x}+2)(\sqrt[3]{3 x}-2)$$

14. $$(\sqrt[3]{4 x}+3)(\sqrt[3]{4 x}-3)$$
Contestar

1.

1. $$14+5 \sqrt{7}$$

2. $$4 \sqrt[3]{6}+3 \sqrt[3]{4}$$

3.

3.

1. $$44-3 \sqrt{11}$$

2. $$3 \sqrt[4]{2}+\sqrt[4]{54}$$

3.

5. $$60+2 \sqrt{3}$$

7.

1. $$30+18 \sqrt{2}$$

2. $$\sqrt[3]{x^{2}}-2 \sqrt[3]{x}-3$$

3.

9.

1. $$-54+13 \sqrt{10}$$

2. $$2 \sqrt[3]{x^{2}}+8 \sqrt[3]{x}+6$$

3.

11. $$23+3 \sqrt{30}$$

13. $$-439-2 \sqrt{77}$$

15.

1. $$14+6 \sqrt{5}$$

2. $$79-20 \sqrt{3}$$

17.

1. $$87-18 \sqrt{6}$$

2. $$163+60 \sqrt{7}$$

19. $$14$$

21. $$-227$$

23. $$19$$

25. $$\sqrt[3]{9 x^{2}}-4$$

##### Ejercicio D: práctica mixta
1. $$\frac{2}{3} \sqrt{27}+\frac{3}{4} \sqrt{48}$$

2. $$\sqrt{175 k^{4}}-\sqrt{63 k^{4}}$$

3. $$\frac{5}{6} \sqrt{162}+\frac{3}{16} \sqrt{128}$$

4. $$\sqrt[3]{24}+\sqrt[3]{ 81}$$

5. $$\frac{1}{2} \sqrt[4]{80}-\frac{2}{3} \sqrt[4]{405}$$

6. $$8 \sqrt[4]{13}-4 \sqrt[4]{13}-3 \sqrt[4]{13}$$

7. $$5 \sqrt{12 c^{4}}-3 \sqrt{27 c^{6}}$$

8. $$\sqrt{80 a^{5}}-\sqrt{45 a^{5}}$$

9. $$\frac{3}{5} \sqrt{75}-\frac{1}{4} \sqrt{48}$$

10. $$21 \sqrt[3]{9}-2 \sqrt[3]{9}$$

11. $$8 \sqrt[3]{64 q^{6}}-3 \sqrt[3]{125 q^{6}}$$

12. $$11 \sqrt{11}-10 \sqrt{11}$$

13. $$\sqrt{3} \cdot \sqrt{21}$$

14. $$(4 \sqrt{6})(-\sqrt{18})$$

15. $$(7 \sqrt[3]{4})(-3 \sqrt[3]{18})$$

16. $$\left(4 \sqrt{12 x^{5}}\right)\left(2 \sqrt{6 x^{3}}\right)$$

17. $$(\sqrt{29})^{2}$$

18. $$(-4 \sqrt{17})(-3 \sqrt{17})$$

19. $$(-4+\sqrt{17})(-3+\sqrt{17})$$

20. $$\left(3 \sqrt[4]{8 a^{2}}\right)\left(\sqrt[4]{12 a^{3}}\right)$$

21. $$(6-3 \sqrt{2})^{2}$$

22. $$\sqrt{3}(4-3 \sqrt{3})$$

23. $$\sqrt[3]{3}(2 \sqrt[3]{9}+\sqrt[3]{18})$$

24. $$(\sqrt{6}+\sqrt{3})(\sqrt{6}+6 \sqrt{3})$$
Contestar

1. $$5\sqrt{3}$$

3. $$9\sqrt{2}$$

5. $$-\sqrt[4]{5}$$

7. $$10 c^{2} \sqrt{3}-9 c^{3} \sqrt{3}$$

9. $$2 \sqrt{3}$$

11. $$17 q^{2}$$

13. $$3 \sqrt{7}$$

15. $$-42 \sqrt[3]{9}$$

17. $$29$$

19. $$29-7 \sqrt{17}$$

21. $$72-36 \sqrt{2}$$

23. $$6+3 \sqrt[3]{2}$$

##### Ejercicio E: ejercicios de escritura
1. Explicar cuando una expresión radical está en forma más simple.
2. Explicar el proceso para determinar si dos radicales son similares o diferentes. Asegúrate de que tu respuesta tenga sentido para los radicales que contienen tanto números como variables.
1. Explicar por qué siempre $$(-\sqrt{n})^{2}$$ es no negativo, para $$n \geq 0$$.
2. Explicar por qué siempre $$-(\sqrt{n})^{2}$$ es no positivo, para $$n \geq 0$$.
3. Utilice el patrón cuadrado binomial para simplificar $$(3+\sqrt{2})^{2}$$. Explica todos tus pasos.
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. En una escala de 1-10, ¿cómo calificaría su dominio de esta sección a la luz de sus respuestas en la lista de verificación? ¿Cómo se puede mejorar esto?

This page titled 8.5E: Ejercicios 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 the style and standards of the LibreTexts platform.