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# 8.3E: Ejercicios

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### La práctica hace perfecto

##### Ejercicio SET A: utilizar la propiedad del producto 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. $$\sqrt[3]{81}$$
Responder

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 del producto 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. $$\dfrac{15+\sqrt{75}}{5}$$
Responder

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 es real

15.

1. $$-2$$
2. no es 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

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. $$\sqrt[4]{\dfrac{160 r^{10}}{5 r^{3}}}$$
Responder

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. Explica por qué $$\sqrt{x^{4}}=x^{2}$$. Entonces explica por qué $$\sqrt{x^{16}}=x^{8}$$.
2. Explica por qué no $$7+\sqrt{9}$$ es igual a $$\sqrt{7+9}$$.
3. Explica cómo sabes eso $$\sqrt[5]{x^{10}}=x^{2}$$.
4. Explica por qué no $$\sqrt[4]{-64}$$ es un número real pero sí lo $$\sqrt[3]{-64}$$ es.
Responder

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 hará para tener confianza en todos los objetivos?

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