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\( \begin{array}{l}\text { Aplica las propledades de.la radicación para simplificar } \\ \text { los slgulentes radicales. } \\ \text { 25. } \sqrt{(-7)^{2}}\end{array} \) 33. \( \left.3 \sqrt[3]{-27 y^{-6} x^{9}}\right) ~ \begin{array}{ll}\text { 26. } \sqrt[5]{15.625 m^{10} n^{15}} & \text { 34. } 10 \sqrt{64 x^{10} y^{8} z^{6}} \\ \text { 27. } \sqrt[3]{\frac{27 b^{12}}{y^{8}}} & \text { 35. } \frac{\sqrt{12 a^{4} b^{6}}}{\sqrt{3 b^{2}}} \\ \text { 28. } \frac{\sqrt[4]{256 \sqrt{256 n^{8}}}}{\sqrt{n^{36}}} & \text { 36. } \sqrt[4]{\sqrt{\frac{1}{512^{16} y^{24} z^{32}}}} \\ \text { 29. } \sqrt{x^{2} \sqrt[3]{64 a^{15} y^{6}}} & \text { 37. } 4 \sqrt[5]{-243\left(m^{9} n^{3}\right)^{20}} \\ \text { 30. } \sqrt[3]{-729 p^{6} q^{9}} & \text { 38. } \sqrt[3]{\sqrt[3]{64 a^{24} b^{30} c^{216}}} \\ \text { 32. } \frac{\sqrt[4]{a^{8} \sqrt{b^{12} c^{16} n^{40}}}}{a^{8} b^{10} \sqrt{n^{20} c^{30}}} & \text { 39. } \sqrt[4]{\sqrt{\left(\frac{y^{51} z^{4}}{a b}\right) \cdot \frac{a^{9}}{y^{3} z}}}\end{array} \)

Ask by Hardy Weston. in Colombia
Feb 27,2025

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Aquí tienes la simplificación de los radicales proporcionados: 1. \( \sqrt{(-7)^{2}} = 7 \) 2. \( \sqrt[5]{15.625 m^{10} n^{15}} = 5^{\frac{4}{5}} \cdot m^{2} \cdot n^{3} \) 3. \( \sqrt[3]{\frac{27 b^{12}}{y^{8}}} = \frac{3 \cdot b^{4}}{y^{\frac{8}{3}}} \) 4. \( \frac{\sqrt[4]{256 \sqrt{256 n^{8}}}}{\sqrt{n^{36}}} = 8n^{-17} \) 5. \( \sqrt{x^{2} \sqrt[3]{64 a^{15} y^{6}}} = 2x a^{\frac{5}{2}} y \) 6. \( \sqrt[3]{-729 p^{6} q^{9}} = -9 p^{2} q^{3} \) 7. \( \frac{\sqrt[4]{a^{8} \sqrt{b^{12} c^{16} n^{40}}}}{a^{8} b^{10} \sqrt{n^{20} c^{30}}} = a^{-6} b^{-\frac{17}{2}} n^{-5} c^{-13} \) 8. \( 3 \sqrt[3]{-27 y^{-6} x^{9}} = -9 x^{3} y^{-2} \) 9. \( 10 \sqrt{64 x^{10} y^{8} z^{6}} = 80 x^{5} y^{4} z^{3} \) 10. \( \frac{\sqrt{12 a^{4} b^{6}}}{\sqrt{3 b^{2}}} = 2 a^{2} b^{2} \) 11. \( \sqrt[4]{\sqrt{\frac{1}{512^{16} y^{24} z^{32}}}} = \frac{1}{2^{36} y^{6} z^{8}} \) 12. \( 4 \sqrt[5]{-243\left(m^{9} n^{3}\right)^{20}} = -12 m^{36} n^{12} \) 13. \( \sqrt[3]{\sqrt[3]{64 a^{24} b^{30} c^{216}}} = \sqrt[3]{4 a^{8} b^{10} c^{72}} \) Estos son los resultados simplificados de cada radical aplicando las propiedades de la radicación.

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Bonus Knowledge

Para simplificar \( \sqrt{(-7)^{2}} \), primero calculamos \((-7)^{2}\), que es \(49\). Luego, aplicamos la raíz cuadrada, obteniendo \( \sqrt{49} = 7\). Dado que estamos trabajando con un cuadrado de un número negativo, la raíz cuadrada regresa el valor positivo. En cuanto a \(3 \sqrt[3]{-27 y^{-6} x^{9}}\), comenzamos simplificando el radical. La raíz cúbica de \(-27\) es \(-3\), ya que \((-3)^{3} = -27\). Para los factores \(y^{-6}\) y \(x^{9}\), tenemos \(\sqrt[3]{y^{-6}} = \frac{1}{y^{2}}\) y \(\sqrt[3]{x^{9}} = x^{3}\). Entonces, al juntar todo, tenemos \(3(-3)(\frac{1}{y^{2}})(x^{3}) = -9 \frac{x^{3}}{y^{2}}\).

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