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# 8.3E: Ejercicios

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### La práctica hace la perfección

##### Ejercicio SET A: utilizar la propiedad product para simplificar expresiones radicales

En los siguientes ejercicios, utilice la Propiedad del Producto para simplificar expresiones radicales.

1. $$\sqrt{27}$$
2. $$\sqrt{80}$$
3. $$\sqrt{125}$$
4. $$\sqrt{96}$$
5. $$\sqrt{147}$$
6. $$\sqrt{450}$$
7. $$\sqrt{800}$$
8. $$\sqrt{675}$$
1. $$\sqrt[4]{32}$$
2. $$\sqrt[5]{64}$$
1. $$\sqrt[3]{625}$$
2. $$\sqrt[6]{128}$$
1. $$\sqrt[5]{64}$$
2. $$\sqrt[3]{256}$$
1. $$\sqrt[4]{3125}$$
2. $$\sqrt[3]{81}$$
Contestar

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

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

5. $$7\sqrt{3}$$

7. $$20\sqrt{2}$$

9.

1. $$2 \sqrt[4]{2}$$
2. $$2 \sqrt[5]{2}$$

11.

1. $$2 \sqrt[5]{2}$$
2. $$4 \sqrt[3]{4}$$
##### Ejercicio SET B: utilizar la propiedad product para simplificar expresiones radicales

En los siguientes ejercicios, simplifique el uso de signos de valor absoluto según sea necesario.

1. $$\sqrt{y^{11}}$$
2. $$\sqrt[3]{r^{5}}$$
3. $$\sqrt[4]{s^{10}}$$
1. $$\sqrt{m^{13}}$$
2. $$\sqrt[5]{u^{7}}$$
3. $$\sqrt[6]{v^{11}}$$
1. $$\sqrt{n^{21}}$$
2. $$\sqrt[3]{q^{8}}$$
3. $$\sqrt[8]{n^{10}}$$
1. $$\sqrt{r^{25}}$$
2. $$\sqrt[5]{p^{8}}$$
3. $$\sqrt[4]{m^{5}}$$
1. $$\sqrt{125 r^{13}}$$
2. $$\sqrt[3]{108 x^{5}}$$
3. $$\sqrt[4]{48 y^{6}}$$
1. $$\sqrt{80 s^{15}}$$
2. $$\sqrt[5]{96 a^{7}}$$
3. $$\sqrt[6]{128 b^{7}}$$
1. $$\sqrt{242 m^{23}}$$
2. $$\sqrt[4]{405 m 10}$$
3. $$\sqrt[5]{160 n^{8}}$$
1. $$\sqrt{175 n^{13}}$$
2. $$\sqrt[5]{512 p^{5}}$$
3. $$\sqrt[4]{324 q^{7}}$$
1. $$\sqrt{147 m^{7} n^{11}}$$
2. $$\sqrt[3]{48 x^{6} y^{7}}$$
3. $$\sqrt[4]{32 x^{5} y^{4}}$$
1. $$\sqrt{96 r^{3} s^{3}}$$
2. $$\sqrt[3]{80 x^{7} y^{6}}$$
3. $$\sqrt[4]{80 x^{8} y^{9}}$$
1. $$\sqrt{192 q^{3} r^{7}}$$
2. $$\sqrt[3]{54 m^{9} n^{10}}$$
3. $$\sqrt[4]{81 a^{9} b^{8}}$$
1. $$\sqrt{150 m^{9} n^{3}}$$
2. $$\sqrt[3]{81 p^{7} q^{8}}$$
3. $$\sqrt[4]{162 c^{11} d^{12}}$$
1. $$\sqrt[3]{-864}$$
2. $$\sqrt[4]{-256}$$
1. $$\sqrt[5]{-486}$$
2. $$\sqrt[6]{-64}$$
1. $$\sqrt[5]{-32}$$
2. $$\sqrt[8]{-1}$$
1. $$\sqrt[3]{-8}$$
2. $$\sqrt[4]{-16}$$
1. $$5+\sqrt{12}$$
2. $$\dfrac{10-\sqrt{24}}{2}$$
1. $$8+\sqrt{96}$$
2. $$\dfrac{8-\sqrt{80}}{4}$$
1. $$1+\sqrt{45}$$
2. $$\dfrac{3+\sqrt{90}}{3}$$
1. $$3+\sqrt{125}$$
2. $$\dfrac{15+\sqrt{75}}{5}$$
Contestar

1.

1. $$\left|y^{5}\right| \sqrt{y}$$
2. $$r \sqrt[3]{r^{2}}$$
3. $$s^{2} \sqrt[4]{s^{2}}$$

3.

1. $$n^{10} \sqrt{n}$$
2. $$q^{2} \sqrt[3]{q^{2}}$$
3. $$|n| \sqrt[8]{n^{2}}$$

5.

1. $$5 r^{6} \sqrt{5 r}$$
2. $$3 x \sqrt[3]{4 x^{2}}$$
3. $$2|y| \sqrt[4]{3 y^{2}}$$

7.

1. $$11\left|m^{11}\right| \sqrt{2 m}$$
2. $$3 m^{2} \sqrt[4]{5 m^{2}}$$
3. $$2 n \sqrt[5]{5 n^{3}}$$

9.

1. $$7\left|m^{3} n^{5}\right| \sqrt{3 m n}$$
2. $$2 x^{2} y^{2} \sqrt[3]{6 y}$$
3. $$2|x y| \sqrt[4]{2 x}$$

11.

1. $$8\left|q r^{3}\right| \sqrt{3 q r}$$
2. $$3 m^{3} n^{3} \sqrt[3]{2 n}$$
3. $$3 a^{2} b^{2} \sqrt[4]{a}$$

13.

1. $$-6 \sqrt[3]{4}$$
2. no real

15.

1. $$-2$$
2. no real

17.

1. $$5+2 \sqrt{3}$$
2. $$5-\sqrt{6}$$

19.

1. $$1+3 \sqrt{5}$$
2. $$1+\sqrt{10}$$
##### Conjunto de ejercicios C: utilizar la propiedad cociente para simplificar expresiones radicales

En los siguientes ejercicios, utilice la Propiedad Cociente para simplificar las raíces cuadradas.

1. $$\sqrt{\dfrac{45}{80}}$$
2. $$\sqrt[3]{\dfrac{8}{27}}$$
3. $$\sqrt[4]{\dfrac{1}{81}}$$
1. $$\sqrt{\dfrac{72}{98}}$$
2. $$\sqrt[3]{\dfrac{24}{81}}$$
3. $$\sqrt[4]{\dfrac{6}{96}}$$
1. $$\sqrt{\dfrac{100}{36}}$$
2. $$\sqrt[3]{\dfrac{81}{375}}$$
3. $$\sqrt[4]{\dfrac{1}{256}}$$
1. $$\sqrt{\dfrac{121}{16}}$$
2. $$\sqrt[3]{\dfrac{16}{250}}$$
3. $$\sqrt[4]{\dfrac{32}{162}}$$
1. $$\sqrt{\dfrac{x^{10}}{x^{6}}}$$
2. $$\sqrt[3]{\dfrac{p^{11}}{p^{2}}}$$
3. $$\sqrt[4]{\dfrac{q^{17}}{q^{13}}}$$
1. $$\sqrt{\dfrac{p^{20}}{p^{10}}}$$
2. $$\sqrt[5]{\dfrac{d^{12}}{d^{7}}}$$
3. $$\sqrt[8]{\dfrac{m^{12}}{m^{4}}}$$
1. $$\sqrt{\dfrac{y^{4}}{y^{8}}}$$
2. $$\sqrt[5]{\dfrac{u^{21}}{u^{11}}}$$
3. $$\sqrt[6]{\dfrac{v^{30}}{v^{12}}}$$
1. $$\sqrt{\dfrac{q^{8}}{q^{14}}}$$
2. $$\sqrt[3]{\dfrac{r^{14}}{r^{5}}}$$
3. $$\sqrt[4]{\dfrac{c^{21}}{c^{9}}}$$
1. $$\sqrt{\dfrac{96 x^{7}}{121}}$$
2. $$\sqrt{\dfrac{108 y^{4}}{49}}$$
3. $$\sqrt{\dfrac{300 m^{5}}{64}}$$
4. $$\sqrt{\dfrac{125 n^{7}}{169}}$$
5. $$\sqrt{\dfrac{98 r^{5}}{100}}$$
6. $$\sqrt{\dfrac{180 s^{10}}{144}}$$
7. $$\sqrt{\dfrac{28 q^{6}}{225}}$$
8. $$\sqrt{\dfrac{150 r^{3}}{256}}$$
1. $$\sqrt{\dfrac{75 r^{9}}{s^{8}}}$$
2. $$\sqrt[3]{\dfrac{54 a^{8}}{b^{3}}}$$
3. $$\sqrt[4]{\dfrac{64 c^{5}}{d^{4}}}$$
1. $$\sqrt{\dfrac{72 x^{5}}{y^{6}}}$$
2. $$\sqrt[5]{\dfrac{96 r^{11}}{s^{5}}}$$
3. $$\sqrt[6]{\dfrac{128 u^{7}}{v^{12}}}$$
1. $$\sqrt{\dfrac{28 p^{7}}{q^{2}}}$$
2. $$\sqrt[3]{\dfrac{81 s^{8}}{t^{3}}}$$
3. $$\sqrt[4]{\dfrac{64 p^{15}}{q^{12}}}$$
1. $$\sqrt{\dfrac{45 r^{3}}{s^{10}}}$$
2. $$\sqrt[3]{\dfrac{625 u^{10}}{v^{3}}}$$
3. $$\sqrt[4]{\dfrac{729 c^{21}}{d^{8}}}$$
1. $$\sqrt{\dfrac{32 x^{5} y^{3}}{18 x^{3} y}}$$
2. $$\sqrt[3]{\dfrac{5 x^{6} y^{9}}{40 x^{5} y^{3}}}$$
3. $$\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}$$
1. $$\sqrt{\dfrac{75 r^{6} s^{8}}{48 r s^{4}}}$$
2. $$\sqrt[3]{\dfrac{24 x^{8} y^{4}}{81 x^{2} y}}$$
3. $$\sqrt[4]{\dfrac{32 m^{9} n^{2}}{162 m n^{2}}}$$
1. $$\sqrt{\dfrac{27 p^{2} q}{108 p^{4} q^{3}}}$$
2. $$\sqrt[3]{\dfrac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$$
3. $$\sqrt[6]{\dfrac{2 m^{9} n^{7}}{128 m^{3} n}}$$
1. $$\sqrt{\dfrac{50 r^{5} s^{2}}{128 r^{2} s^{6}}}$$
2. $$\sqrt[3]{\dfrac{24 m^{9} n^{7}}{375 m^{4} n}}$$
3. $$\sqrt[4]{\dfrac{81 m^{2} n^{8}}{256 m^{1} n^{2}}}$$
1. $$\dfrac{\sqrt{45 p^{9}}}{\sqrt{5 q^{2}}}$$
2. $$\dfrac{\sqrt[4]{64}}{\sqrt[4]{2}}$$
3. $$\dfrac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$$
1. $$\dfrac{\sqrt{80 q^{5}}}{\sqrt{5 q}}$$
2. $$\dfrac{\sqrt[3]{-625}}{\sqrt[3]{5}}$$
3. $$\dfrac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}$$
1. $$\dfrac{\sqrt{50 m^{7}}}{\sqrt{2 m}}$$
2. $$\sqrt[3]{\dfrac{1250}{2}}$$
3. $$\sqrt[4]{\dfrac{486 y^{9}}{2 y^{3}}}$$
1. $$\dfrac{\sqrt{72 n^{11}}}{\sqrt{2 n}}$$
2. $$\sqrt[3]{\dfrac{162}{6}}$$
3. $$\sqrt[4]{\dfrac{160 r^{10}}{5 r^{3}}}$$
Contestar

1.

1. $$\dfrac{3}{4}$$
2. $$\dfrac{2}{3}$$
3. $$\dfrac{1}{3}$$

3.

1. $$\dfrac{5}{3}$$
2. $$\dfrac{3}{5}$$
3. $$\dfrac{1}{4}$$

5.

1. $$x^{2}$$
2. $$p^{3}$$
3. $$|q|$$

7.

1. $$\dfrac{1}{y^{2}}$$
2. $$u^{2}$$
3. $$|v^{3}|$$

9. $$\dfrac{4\left|x^{3}\right| \sqrt{6 x}}{11}$$

11. $$\dfrac{10 m^{2} \sqrt{3 m}}{8}$$

13. $$\dfrac{7 r^{2} \sqrt{2 r}}{10}$$

15. $$\dfrac{2\left|q^{3}\right| \sqrt{7}}{15}$$

17.

1. $$\dfrac{5 r^{4} \sqrt{3 r}}{s^{4}}$$
2. $$\dfrac{3 a^{2} \sqrt[3]{2 a^{2}}}{|b|}$$
3. $$\dfrac{2|c| \sqrt[4]{4 c}}{|d|}$$

19.

1. $$\dfrac{2\left|p^{3}\right| \sqrt{7 p}}{|q|}$$
2. $$\dfrac{3 s^{2} \sqrt[3]{3 s^{2}}}{t}$$
3. $$\dfrac{2\left|p^{3}\right| \sqrt[4]{4 p^{3}}}{\left|q^{3}\right|}$$

21.

1. $$\dfrac{4|x y|}{3}$$
2. $$\dfrac{y^{2} \sqrt[3]{x}}{2}$$
3. $$\dfrac{|a b| \sqrt[4]{a}}{4}$$

23.

1. $$\dfrac{1}{2|p q|}$$
2. $$\dfrac{2 c d \sqrt[5]{2 d^{2}}}{5}$$
3. $$\dfrac{|m n| \sqrt[6]{2}}{2}$$

25.

1. $$\dfrac{3 p^{4} \sqrt{p}}{|q|}$$
2. $$2 \sqrt[4]{2}$$
3. $$2 x \sqrt[5]{2 x}$$

27.

1. $$5\left|m^{3}\right|$$
2. $$5 \sqrt[3]{5}$$
3. $$3|y| \sqrt[4]{3 y^{2}}$$
##### Ejercicio SET D: ejercicios de escritura
1. Explique por qué$$\sqrt{x^{4}}=x^{2}$$. Entonces explica por qué$$\sqrt{x^{16}}=x^{8}$$.
2. $$7+\sqrt{9}$$Explique por qué no es igual a$$\sqrt{7+9}$$.
3. Explica cómo lo sabes$$\sqrt[5]{x^{10}}=x^{2}$$.
4. $$\sqrt[4]{-64}$$Explique por qué no es un número real sino$$\sqrt[3]{-64}$$ que sí.
Contestar

1. Las respuestas pueden variar

3. Las respuestas pueden variar

## 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 revisar esta lista de verificación, ¿qué harás para tener confianza en todos los objetivos?

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