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# 8.4E: Ejercicios

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### La práctica hace la perfección

##### Ejercicio SET A: simplificar expresiones con$$a^{\frac{1}{n}}$$

En los siguientes ejercicios, escribe como expresión radical.

1. a.$$x^{\frac{1}{2}}$$ b.$$y^{\frac{1}{3}}$$ c.$$z^{\frac{1}{4}}$$
2. a.$$r^{\frac{1}{2}}$$ b.$$s^{\frac{1}{3}}$$ c.$$t^{\frac{1}{4}}$$
3. a.$$u^{\frac{1}{5}}$$ b.$$v^{\frac{1}{9}}$$ c.$$w^{\frac{1}{20}}$$
4. a.$$g^{\frac{1}{7}}$$ b.$$h^{\frac{1}{5}}$$ c.$$j^{\frac{1}{25}}$$
Contestar

1. a.$$\sqrt{x}$$ b.$$\sqrt[3]{y}$$ c.$$\sqrt[4]{z}$$

3. a.$$\sqrt[5]{u}$$ b.$$\sqrt[9]{v}$$ c.$$\sqrt[20]{w}$$

##### Conjunto de ejercicios B: simplificar expresiones con$$a^{\frac{1}{n}}$$

En los siguientes ejercicios, escribe con un exponente racional.

1. a.$$\sqrt[7]{x}$$ b.$$\sqrt[9]{y}$$ c.$$\sqrt[5]{f}$$
2. a.$$\sqrt[8]{4}$$ b.$$\sqrt[10]{s}$$ c.$$\sqrt[4]{t}$$
3. a.$$\sqrt[3]{7c}$$ b.$$\sqrt[7]{12d}$$ c.$$2\sqrt[4]{6b}$$
4. a.$$\sqrt[4]{5x}$$ b.$$\sqrt[8]{9y}$$ c.$$7\sqrt[5]{3z}$$
5. a.$$\sqrt{21p}$$ b.$$\sqrt[4]{8q}$$ c.$$4\sqrt[6]{36r}$$
6. a.$$\sqrt[3]{25a}$$ b.$$\sqrt{3b}$$ c.$$\sqrt[8]{40c}$$
Contestar

1. a.$$x^{\frac{1}{7}}$$ b.$$y^{\frac{1}{9}}$$ c.$$f^{\frac{1}{5}}$$

3. a.$$(7 c)^{\frac{1}{4}}$$ b.$$(12 d)^{\frac{1}{7}}$$ c.$$2(6 b)^{\frac{1}{4}}$$

5. a.$$(21 p)^{\frac{1}{2}}$$ b.$$(8 q)^{\frac{1}{4}}$$ c.$$4(36 r)^{\frac{1}{6}}$$

##### Ejercicio SET C: simplificar expresiones con$$a^{\frac{1}{n}}$$

En los siguientes ejercicios, simplifique.

1. a.$$81^{\frac{1}{2}}$$ b.$$125^{\frac{1}{3}}$$ c.$$64^{\frac{1}{2}}$$
2. a.$$625^{\frac{1}{4}}$$ b.$$243^{\frac{1}{5}}$$ c.$$32^{\frac{1}{5}}$$
3. a.$$16^{\frac{1}{4}}$$ b.$$16^{\frac{1}{2}}$$ c.$$625^{\frac{1}{4}}$$
4. a.$$64^{\frac{1}{3}}$$ b.$$32^{\frac{1}{5}}$$ c.$$81^{\frac{1}{4}}$$
5. a.$$(-216)^{\frac{1}{3}}$$ b.$$-216^{\frac{1}{3}}$$ c.$$(216)^{-\frac{1}{3}}$$
6. a.$$(-1000)^{\frac{1}{3}}$$ b.$$-1000^{\frac{1}{3}}$$ c.$$(1000)^{-\frac{1}{3}}$$
7. a.$$(-81)^{\frac{1}{4}}$$ b.$$-81^{\frac{1}{4}}$$ c.$$(81)^{-\frac{1}{4}}$$
8. a.$$(-49)^{\frac{1}{2}}$$ b.$$-49^{\frac{1}{2}}$$ c.$$(49)^{-\frac{1}{2}}$$
9. a.$$(-36)^{\frac{1}{2}}$$ b.$$-36^{\frac{1}{2}}$$ c.$$(36)^{-\frac{1}{2}}$$
10. a.$$(-16)^{\frac{1}{4}}$$ b.$$-16^{\frac{1}{4}}$$ c.$$16^{-\frac{1}{4}}$$
11. a.$$(-100)^{\frac{1}{2}}$$ b.$$-100^{\frac{1}{2}}$$ c.$$(100)^{-\frac{1}{2}}$$
12. a.$$(-32)^{\frac{1}{5}}$$ b.$$(243)^{-\frac{1}{5}}$$ c.$$-125^{\frac{1}{3}}$$
Contestar

1. a.$$9$$ b.$$5$$ c.$$8$$

3. a.$$2$$ b.$$4$$ c.$$5$$

5. a.$$-6$$ b.$$-6$$ c.$$\frac{1}{6}$$

7. a. no real b.$$-3$$ c.$$\frac{1}{3}$$

9. a. no real b.$$-6$$ c.$$\frac{1}{6}$$

11. a. no real b.$$-10$$ c.$$\frac{1}{10}$$

##### Ejercicio SET D: simplificar expresiones con$$a^{\frac{m}{n}}$$

En los siguientes ejercicios, escribe con un exponente racional.

1. a.$$\sqrt{m^{5}}$$ b.$$(\sqrt[3]{3 y})^{7}$$ c.$$\sqrt[5]{\left(\dfrac{4 x}{5 y}\right)^{3}}$$
2. a.$$\sqrt[4]{r^{7}}$$ b.$$(\sqrt[5]{2 p q})^{3}$$ c.$$\sqrt[4]{\left(\dfrac{12 m}{7 n}\right)^{3}}$$
3. a.$$\sqrt[5]{u^{2}}$$ b.$$(\sqrt[3]{6 x})^{5}$$ c.$$\sqrt[4]{\left(\dfrac{18 a}{5 b}\right)^{7}}$$
4. a.$$\sqrt[3]{a}$$ b.$$(\sqrt[4]{21 v})^{3}$$ c.$$\sqrt[4]{\left(\dfrac{2 x y}{5 z}\right)^{2}}$$
Contestar

1. a.$$m^{\frac{5}{2}}$$ b.$$(3 y)^{\frac{7}{3}}$$ c.$$\left(\dfrac{4 x}{5 y}\right)^{\frac{3}{5}}$$

3. a.$$u^{\frac{2}{5}}$$ b.$$(6 x)^{\frac{5}{3}}$$ c.$$\left(\dfrac{18 a}{5 b}\right)^{\frac{7}{4}}$$

##### Ejercicio SET E: simplificar expresiones con$$a^{\frac{m}{n}}$$

En los siguientes ejercicios, simplifique.

1. a.$$64^{\frac{5}{2}}$$ b.$$81^{\frac{-3}{2}}$$ c.$$(-27)^{\frac{2}{3}}$$
2. a.$$25^{\frac{3}{2}}$$ b.$$9^{-\frac{3}{2}}$$ c.$$(-64)^{\frac{2}{3}}$$
3. a.$$32^{\frac{2}{5}}$$ b.$$27^{-\frac{2}{3}}$$ c.$$(-25)^{\frac{1}{2}}$$
4. a.$$100^{\frac{3}{2}}$$ b.$$49^{-\frac{5}{2}}$$ c.$$(-100)^{\frac{3}{2}}$$
5. a.$$-9^{\frac{3}{2}}$$ b.$$-9^{-\frac{3}{2}}$$ c.$$(-9)^{\frac{3}{2}}$$
6. a.$$-64^{\frac{3}{2}}$$ b.$$-64^{-\frac{3}{2}}$$ c.$$(-64)^{\frac{3}{2}}$$
Contestar

1. a.$$32,768$$ b.$$\frac{1}{729}$$ c.$$9$$

3. a.$$4$$ b.$$\frac{1}{9}$$ c. no real

5. a.$$-27$$ b.$$-\frac{1}{27}$$ c. no real

##### Ejercicio SET F: usar las leyes de los exponentes para simplificar expresiones con exponentes racionales

En los siguientes ejercicios, simplifique. Supongamos que todas las variables son positivas.

1. a.$$c^{\frac{1}{4}} \cdot c^{\frac{5}{8}}$$ b.$$\left(p^{12}\right)^{\frac{3}{4}}$$ c.$$\dfrac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}$$
2. a.$$6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$$ b.$$\left(b^{15}\right)^{\frac{3}{5}}$$ c.$$\dfrac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$$
3. a.$$y^{\frac{1}{2}} \cdot y^{\frac{3}{4}}$$ b.$$\left(x^{12}\right)^{\frac{2}{3}}$$ c.$$\dfrac{m^{\frac{5}{8}}}{m^{\frac{13}{8}}}$$
4. a.$$q^{\frac{2}{3}} \cdot q^{\frac{5}{6}}$$ b.$$\left(h^{6}\right)^{\frac{4}{3}}$$ c.$$\dfrac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}$$
5. a.$$\left(27 q^{\frac{3}{2}}\right)^{\frac{4}{3}}$$ b.$$\left(a^{\frac{1}{3}} b^{\frac{2}{3}}\right)^{\frac{3}{2}}$$
6. a.$$\left(64 s^{\frac{3}{7}}\right)^{\frac{1}{6}}$$ b.$$\left(m^{\frac{4}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{4}}$$
7. a.$$\left(16 u^{\frac{1}{3}}\right)^{\frac{3}{4}}$$ b.$$\left(4 p^{\frac{1}{3}} q^{\frac{1}{2}}\right)^{\frac{3}{2}}$$
8. a.$$\left(625 n^{\frac{8}{3}}\right)^{\frac{3}{4}}$$ b.$$\left(9 x^{\frac{2}{5}} y^{\frac{3}{5}}\right)^{\frac{5}{2}}$$
9. a.$$\dfrac{r^{\frac{5}{2}} \cdot r^{-\frac{1}{2}}}{r^{-\frac{3}{2}}}$$ b.$$\left(\dfrac{36 s^{\frac{1}{5}} t^{-\frac{3}{2}}}{s^{-\frac{9}{5}} t^{\frac{1}{2}}}\right)^{\frac{1}{2}}$$
10. a.$$\dfrac{a^{\frac{3}{4}} \cdot a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}}$$ b.$$\left(\dfrac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}}\right)^{\frac{1}{3}}$$
11. a.$$\dfrac{c^{\frac{5}{3}} \cdot c^{-\frac{1}{3}}}{c^{-\frac{2}{3}}}$$ b.$$\left(\dfrac{8 x^{\frac{5}{3}} y^{-\frac{1}{2}}}{27 x^{-\frac{4}{3}} y^{\frac{5}{2}}}\right)^{\frac{1}{3}}$$
12. a.$$\dfrac{m^{\frac{7}{4}} \cdot m^{-\frac{5}{4}}}{m^{-\frac{2}{4}}}$$ b.$$\left(\dfrac{16 m^{\frac{1}{5}} n^{\frac{3}{2}}}{81 m^{\frac{9}{5}} n^{-\frac{1}{2}}}\right)^{\frac{1}{4}}$$
Contestar

1. a.$$c^{\frac{7}{8}}$$ b.$$p^{9}$$ c.$$\frac{1}{r}$$

3. a.$$y^{\frac{5}{4}}$$ b.$$x^{8}$$ c.$$\dfrac{1}{m}$$

5. a.$$81 q^{2}$$ b.$$a^{\frac{1}{2}} b$$

7. a.$$8 u^{\frac{1}{4}}$$ b.$$8 p^{\frac{1}{2}} q^{\frac{3}{4}}$$

9. a.$$r^{\frac{7}{2}}$$ b.$$\dfrac{6 s}{t}$$

11. a.$$c^{2}$$ b.$$\dfrac{2x}{3y}$$

##### Ejercicio SET G: ejercicios de escritura
1. Mostrar dos métodos algebraicos diferentes para simplificar$$4^{\frac{3}{2}}$$. Explica todos tus pasos.
2. Explique por qué la expresión$$(-16)^{\frac{3}{2}}$$ no puede ser evaluada.
Contestar

1. 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. ¿Qué te dice esta lista de verificación sobre tu dominio de esta sección? ¿Qué pasos tomarás para mejorar?

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