(1) RAZONAMIENTO. Cれícular las siguientes raices aplicando las propiedades de losiradicales: 13. \( \sqrt{(-5)^{2}} \) 14. \( \sqrt[3]{-729 x^{6}} \) 15. \( \sqrt[4]{16 a^{8} b^{4}} \) 16. \( \sqrt[5]{3.125 m^{10} n^{5}} \) 17. \( \sqrt[2]{64 x^{10} y^{8} z^{6}} \) 18. \( \sqrt[3]{-27 x^{-6} y^{9}} \) 19. \( \sqrt{\frac{9 y^{8}}{m^{12}}} \) 20. \( \frac{\sqrt{12 a^{4} b^{6}}}{1 \sqrt{3 b^{2}}} \) \( : 27 \cdot \sqrt[5]{\frac{128 x^{15} y^{20}}{4 z^{10}}} \) 22. \( \sqrt[3]{4 \sqrt{4 m^{6}}} \) 23. \( \sqrt[4]{\sqrt[2]{m^{16} y^{24} z^{8}}} \) 24. \( \sqrt{x^{2} \sqrt[3]{\frac{1}{64 y^{6}}}} \) - MODELACIÓN: Escribir las siguientes raíces en forma de potencia y simplificar cada resultado si es posible. 25. \( \sqrt[5]{x^{5} y^{10}} \) 26. \( \sqrt[3]{w^{3} z^{3}} \) 27. \( \sqrt{4 y^{4}} \) 28. \( \sqrt[6]{64 x^{12} y^{6}} \) 29. \( \sqrt[n]{m^{2 n^{n}}} \) 30. \( \sqrt[4]{7^{8} x^{16} y^{8}} \) 31. \( \sqrt[10]{10^{10} x_{i}^{20}} \) 32. \( \sqrt[3]{27 x^{3}} \) 33. \( \sqrt[5]{243 x^{10}} \)

Simplify the expression by following steps: - step0: Solution: \(\sqrt{4y^{4}}\) - step1: Transform the expression: \(\sqrt{\left(2y^{2}\right)^{2}}\) - step2: Simplify the root: \(2y^{2}\) Calculate or simplify the expression \( \sqrt[5]{243 * x^{10}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[5]{243x^{10}}\) - step1: Transform the expression: \(\sqrt[5]{\left(3x^{2}\right)^{5}}\) - step2: Simplify the root: \(3x^{2}\) Calculate or simplify the expression \( \sqrt[3]{4 * \sqrt(4 * m^6)} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[3]{4\sqrt{4m^{6}}}\) - step1: Simplify the root: \(\sqrt[3]{4\times 2m^{3}}\) - step2: Multiply the numbers: \(\sqrt[3]{8m^{3}}\) - step3: Transform the expression: \(\sqrt[3]{\left(2m\right)^{3}}\) - step4: Simplify the root: \(2m\) Calculate or simplify the expression \( \sqrt[4]{7^8 * x^{16} * y^8} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[4]{7^{8}x^{16}y^{8}}\) - step1: Transform the expression: \(\sqrt[4]{\left(7^{2}x^{4}y^{2}\right)^{4}}\) - step2: Simplify the root: \(7^{2}x^{4}y^{2}\) - step3: Calculate: \(49x^{4}y^{2}\) Calculate or simplify the expression \( \sqrt[6]{64 * x^{12} * y^6} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[6]{64x^{12}y^{6}}\) - step1: Transform the expression: \(\sqrt[6]{\left(2x^{2}y\right)^{6}}\) - step2: Simplify the root: \(2x^{2}y\) Calculate or simplify the expression \( \sqrt[n]{m^{2 * n^n}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[n]{m^{2n^{n}}}\) - step1: Rewrite the expression: \(m^{2n^{n}\times \frac{1}{n}}\) - step2: Multiply the terms: \(m^{2n^{n-1}}\) Calculate or simplify the expression \( \sqrt[10]{10^{10} * x_i^{20}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[10]{10^{10}x_{i}^{20}}\) - step1: Transform the expression: \(\sqrt[10]{\left(10x_{i}^{2}\right)^{10}}\) - step2: Simplify the root: \(10x_{i}^{2}\) Calculate or simplify the expression \( \sqrt[2]{64 * x^{10} * y^{8} * z^{6}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{64x^{10}y^{8}z^{6}}\) - step1: Transform the expression: \(\sqrt{\left(8x^{5}y^{4}z^{3}\right)^{2}}\) - step2: Simplify the root: \(8x^{5}y^{4}z^{3}\) Calculate or simplify the expression \( \sqrt((-5)^2) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{\left(-5\right)^{2}}\) - step1: Simplify the root: \(\left|-5\right|\) - step2: Calculate the absolute value: \(5\) Calculate or simplify the expression \( \sqrt[4]{16 * a^8 * b^4} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[4]{16a^{8}b^{4}}\) - step1: Transform the expression: \(\sqrt[4]{\left(2a^{2}b\right)^{4}}\) - step2: Simplify the root: \(2a^{2}b\) Calculate or simplify the expression \( \sqrt[3]{-729 * x^6} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[3]{-729x^{6}}\) - step1: Use the properties of radicals: \(-\sqrt[3]{729x^{6}}\) - step2: Simplify the expression: \(-9x^{2}\) Calculate or simplify the expression \( \sqrt[5]{3.125 * m^{10} * n^{5}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[5]{3.125m^{10}n^{5}}\) - step1: Rewrite the expression: \(\sqrt[5]{\frac{25m^{10}n^{5}}{8}}\) - step2: Use the properties of radicals: \(\frac{\sqrt[5]{25m^{10}n^{5}}}{\sqrt[5]{8}}\) - step3: Simplify the expression: \(\frac{\sqrt[5]{25}\times nm^{2}}{\sqrt[5]{8}}\) - step4: Simplify: \(\frac{\sqrt[5]{100}\times nm^{2}}{2}\) Calculate or simplify the expression \( \sqrt[4]{\sqrt(m^{16} * y^{24} * z^{8})} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[4]{\sqrt{m^{16}y^{24}z^{8}}}\) - step1: Simplify the expression: \(\sqrt[4\times 2]{m^{16}y^{24}z^{8}}\) - step2: Multiply the numbers: \(\sqrt[8]{m^{16}y^{24}z^{8}}\) - step3: Transform the expression: \(\sqrt[8]{\left(m^{2}y^{3}z\right)^{8}}\) - step4: Simplify the root: \(m^{2}y^{3}z\) Calculate or simplify the expression \( \sqrt(x^2 * \sqrt[3]{1 / (64 * y^6)}) \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{x^{2}\sqrt[3]{\frac{1}{64y^{6}}}}\) - step1: Simplify the root: \(\sqrt{x^{2}\times \frac{1}{4y^{2}}}\) - step2: Multiply the terms: \(\sqrt{\frac{x^{2}}{4y^{2}}}\) - step3: Use the properties of radicals: \(\frac{\sqrt{x^{2}}}{\sqrt{4y^{2}}}\) - step4: Simplify the expression: \(\frac{x}{2y}\) Calculate or simplify the expression \( \sqrt[3]{27 * x^3} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[3]{27x^{3}}\) - step1: Transform the expression: \(\sqrt[3]{\left(3x\right)^{3}}\) - step2: Simplify the root: \(3x\) Calculate or simplify the expression \( \sqrt(9 * y^8 / m^{12}) \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{\frac{9y^{8}}{m^{12}}}\) - step1: Use the properties of radicals: \(\frac{\sqrt{9y^{8}}}{\sqrt{m^{12}}}\) - step2: Simplify the expression: \(\frac{3y^{4}}{m^{6}}\) Calculate or simplify the expression \( \sqrt[3]{-27 * x^{-6} * y^{9}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[3]{-27x^{-6}y^{9}}\) - step1: Rewrite the expression: \(\sqrt[3]{-\frac{27y^{9}}{x^{6}}}\) - step2: Use the properties of radicals: \(-\sqrt[3]{\frac{27y^{9}}{x^{6}}}\) - step3: Simplify the expression: \(-\frac{3y^{3}}{x^{2}}\) Calculate or simplify the expression \( \sqrt[3]{w^3 * z^3} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[3]{w^{3}z^{3}}\) - step1: Transform the expression: \(\sqrt[3]{\left(wz\right)^{3}}\) - step2: Simplify the root: \(wz\) Calculate or simplify the expression \( \sqrt[5]{x^5 * y^{10}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[5]{x^{5}y^{10}}\) - step1: Transform the expression: \(\sqrt[5]{\left(xy^{2}\right)^{5}}\) - step2: Simplify the root: \(xy^{2}\) Calculate or simplify the expression \( 27 * \sqrt[5]{128 * x^{15} * y^{20} / 4 * z^{10}} \). Simplify the expression by following steps: - step0: Solution: \(27\sqrt[5]{\frac{128x^{15}y^{20}}{4}z^{10}}\) - step1: Divide the terms: \(27\sqrt[5]{32x^{15}y^{20}z^{10}}\) - step2: Simplify the root: \(27\times 2x^{3}y^{4}z^{2}\) - step3: Multiply the numbers: \(54x^{3}y^{4}z^{2}\) Calculate or simplify the expression \( \sqrt(12 * a^4 * b^6) / \sqrt(3 * b^2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\sqrt{12a^{4}b^{6}}}{\sqrt{3b^{2}}}\) - step1: Simplify the root: \(\frac{2\sqrt{3}\times b^{3}a^{2}}{\sqrt{3b^{2}}}\) - step2: Simplify the root: \(\frac{2\sqrt{3}\times b^{3}a^{2}}{\sqrt{3}\times b}\) - step3: Reduce the fraction: \(\frac{2\sqrt{3}\times b^{3-1}a^{2}}{\sqrt{3}}\) - step4: Reduce the fraction: \(\frac{2\sqrt{3}\times b^{2}a^{2}}{\sqrt{3}}\) - step5: Reduce the fraction: \(2b^{2}a^{2}\) Aquí están las soluciones a las raíces solicitadas, aplicando las propiedades de los radicales: 13. \( \sqrt{(-5)^{2}} = 5 \) 14. \( \sqrt[3]{-729 x^{6}} = -9x^{2} \) 15. \( \sqrt[4]{16 a^{8} b^{4}} = 2a^{2}b \) 16. \( \sqrt[5]{3.125 m^{10} n^{5}} = \frac{\sqrt[5]{100} \times nm^{2}}{2} \) 17. \( \sqrt[2]{64 x^{10} y^{8} z^{6}} = 8x^{5}y^{4}z^{3} \) 18. \( \sqrt[3]{-27 x^{-6} y^{9}} = -\frac{3y^{3}}{x^{2}} \) 19. \( \sqrt{\frac{9 y^{8}}{m^{12}}} = \frac{3y^{4}}{m^{6}} \) 20. \( \frac{\sqrt{12 a^{4} b^{6}}}{\sqrt{3 b^{2}}} = 2b^{2}a^{2} \) 21. \( 27 \cdot \sqrt[5]{\frac{128 x^{15} y^{20}}{4 z^{10}}} = 54x^{3}y^{4}z^{2} \) 22. \( \sqrt[3]{4 \sqrt{4 m^{6}}} = 2m \) 23. \( \sqrt[4]{\sqrt{m^{16} y^{24} z^{8}}} = m^{2}y^{3}z \) 24. \( \sqrt{x^{2} \sqrt[3]{\frac{1}{64 y^{6}}}} = \frac{x}{2y} \) **MODELACIÓN:** 25. \( \sqrt[5]{x^{5} y^{10}} = xy^{2} \) 26. \( \sqrt[3]{w^{3} z^{3}} = wz \) 27. \( \sqrt{4 y^{4}} = 2y^{2} \) 28. \( \sqrt[6]{64 x^{12} y^{6}} = 2x^{2}y \) 29. \( \sqrt[n]{m^{2 n^{n}}} = m^{2n^{n-1}} \) 30. \( \sqrt[4]{7^{8} x^{16} y^{8}} = 49x^{4}y^{2} \) 31. \( \sqrt[10]{10^{10} x_{i}^{20}} = 10x_{i}^{2} \) 32. \( \sqrt[3]{27 x^{3}} = 3x \) 33. \( \sqrt[5]{243 x^{10}} = 3x^{2} \) Si necesitas más aclaraciones o ejemplos, no dudes en preguntar.