- EIERCITACIÓN: Multiplicar los siguientes radicales. Luego, simplificar si es posible. 18. \( (5 \sqrt{2})(3 \sqrt{12}) \) 19. \( (-5 \sqrt{8})(-6 \sqrt{7}) \) 20. \( \sqrt{5 a b^{7}} \sqrt{10 a^{3} b^{3}} \) 21. \( (9 \sqrt[3]{6 x})\left(2 \sqrt[3]{9 x^{2}}\right) \) 22. \( \sqrt{\frac{3}{4} x^{2} y^{3}} \sqrt{\frac{5}{3} x^{5} y^{-2}} \) 23. \( \sqrt{\frac{3 m n^{2}}{5 c^{4}}} \sqrt{\frac{25 m^{6} n}{12 c^{-4}}} \) \( 24.2 \sqrt{2 a}(3 \sqrt{12 a}-\sqrt{27 a}) \) 25. \( 3 \sqrt[3]{3 a}\left(4 \sqrt[3]{9 a^{2}}+5 \sqrt[3]{7 a^{2}}\right) \) 26. \( (\sqrt{x}+2 \sqrt{y})(2 \sqrt{x}-\sqrt{y}) \) 27. \( (2 \sqrt{x}-3 \sqrt{y})(3 \sqrt{x}+5 \sqrt{y}) \)

Simplify the expression by following steps: - step0: Solution: \(2\sqrt{2a}\times \left(3\sqrt{12a}-\sqrt{27a}\right)\) - step1: Simplify the root: \(2\sqrt{2a}\times \left(3\times 2\sqrt{3a}-\sqrt{27a}\right)\) - step2: Simplify the root: \(2\sqrt{2a}\times \left(3\times 2\sqrt{3a}-3\sqrt{3a}\right)\) - step3: Multiply the terms: \(2\sqrt{2a}\times \left(6\sqrt{3a}-3\sqrt{3a}\right)\) - step4: Subtract the terms: \(2\sqrt{2a}\times 3\sqrt{3a}\) - step5: Multiply the terms: \(6\sqrt{2a}\times \sqrt{3a}\) - step6: Multiply the terms: \(6\sqrt{6}\times a\) Calculate or simplify the expression \( (2 * \sqrt(x) - 3 * \sqrt(y)) * (3 * \sqrt(x) + 5 * \sqrt(y)) \). Simplify the expression by following steps: - step0: Solution: \(\left(2\sqrt{x}-3\sqrt{y}\right)\left(3\sqrt{x}+5\sqrt{y}\right)\) - step1: Use the distributive property: \(2\sqrt{x}\times 3\sqrt{x}+2\sqrt{x}\times 5\sqrt{y}-3\sqrt{y}\times 3\sqrt{x}-3\sqrt{y}\times 5\sqrt{y}\) - step2: Multiply the terms: \(6x+2\sqrt{x}\times 5\sqrt{y}-3\sqrt{y}\times 3\sqrt{x}-3\sqrt{y}\times 5\sqrt{y}\) - step3: Multiply the terms: \(6x+10\sqrt{xy}-3\sqrt{y}\times 3\sqrt{x}-3\sqrt{y}\times 5\sqrt{y}\) - step4: Multiply the terms: \(6x+10\sqrt{xy}-9\sqrt{yx}-3\sqrt{y}\times 5\sqrt{y}\) - step5: Multiply the terms: \(6x+10\sqrt{xy}-9\sqrt{yx}-15y\) Calculate or simplify the expression \( \sqrt((3/4) * x^2 * y^3) * \sqrt((5/3) * x^5 * y^(-2)) \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{\left(\frac{3}{4}\right)x^{2}y^{3}}\times \sqrt{\left(\frac{5}{3}\right)x^{5}y^{-2}}\) - step1: Evaluate: \(\sqrt{\frac{3}{4}x^{2}y^{3}}\times \sqrt{\frac{5}{3}x^{5}y^{-2}}\) - step2: Simplify the root: \(\frac{yx\sqrt{3y}}{2}\sqrt{\frac{5}{3}x^{5}y^{-2}}\) - step3: Simplify the root: \(\frac{yx\sqrt{3y}}{2}\times \frac{x^{2}\sqrt{5x}}{\sqrt{3}\times y}\) - step4: Reduce the fraction: \(\frac{x\sqrt{3y}}{2}\times \frac{x^{2}\sqrt{5x}}{\sqrt{3}}\) - step5: Multiply the terms: \(\frac{x\sqrt{3y}\times x^{2}\sqrt{5x}}{2\sqrt{3}}\) - step6: Multiply the terms: \(\frac{x^{3}\sqrt{15yx}}{2\sqrt{3}}\) - step7: Simplify: \(\frac{x^{3}\sqrt{5xy}}{2}\) Calculate or simplify the expression \( (-5 * \sqrt(8)) * (-6 * \sqrt(7)) \). Calculate the value by following steps: - step0: Calculate: \(\left(-5\sqrt{8}\right)\left(-6\sqrt{7}\right)\) - step1: Remove the parentheses: \(-5\sqrt{8}\times \left(-6\sqrt{7}\right)\) - step2: Rewrite the expression: \(-5\sqrt{8}\times \left(-6\right)\sqrt{7}\) - step3: Simplify the root: \(-5\times 2\sqrt{2}\times \left(-6\right)\sqrt{7}\) - step4: Rewrite the expression: \(5\times 2\sqrt{2}\times 6\sqrt{7}\) - step5: Multiply the terms: \(60\sqrt{2}\times \sqrt{7}\) - step6: Multiply the terms: \(60\sqrt{14}\) Calculate or simplify the expression \( (5 * \sqrt(2)) * (3 * \sqrt(12)) \). Calculate the value by following steps: - step0: Calculate: \(\left(5\sqrt{2}\right)\left(3\sqrt{12}\right)\) - step1: Remove the parentheses: \(5\sqrt{2}\times 3\sqrt{12}\) - step2: Simplify the root: \(5\sqrt{2}\times 3\times 2\sqrt{3}\) - step3: Multiply the terms: \(30\sqrt{2}\times \sqrt{3}\) - step4: Multiply the terms: \(30\sqrt{6}\) Calculate or simplify the expression \( \sqrt(5 * a * b^7) * \sqrt(10 * a^3 * b^3) \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{5ab^{7}}\times \sqrt{10a^{3}b^{3}}\) - step1: Simplify the root: \(b^{3}\sqrt{5ba}\times \sqrt{10a^{3}b^{3}}\) - step2: Simplify the root: \(b^{3}\sqrt{5ba}\times ba\sqrt{10ba}\) - step3: Calculate: \(b^{3}\times ba\times 5\sqrt{2}\times ab\) - step4: Reorder the terms: \(5\sqrt{2}\times b^{3}\times ba\times ab\) - step5: Calculate: \(5\sqrt{2}\times b^{4}a\times ab\) - step6: Calculate: \(5\sqrt{2}\times b^{5}a\times a\) - step7: Multiply the terms: \(5\sqrt{2}\times b^{5}a^{2}\) Calculate or simplify the expression \( (9 * (6 * x)^(1/3)) * (2 * (9 * x^2)^(1/3)) \). Simplify the expression by following steps: - step0: Solution: \(\left(9\left(6x\right)^{\frac{1}{3}}\right)\left(2\left(9x^{2}\right)^{\frac{1}{3}}\right)\) - step1: Remove the parentheses: \(9\left(6x\right)^{\frac{1}{3}}\times 2\left(9x^{2}\right)^{\frac{1}{3}}\) - step2: Multiply the terms: \(18\left(6x\right)^{\frac{1}{3}}\left(9x^{2}\right)^{\frac{1}{3}}\) - step3: Multiply the terms: \(18\times 6^{\frac{1}{3}}x^{\frac{1}{3}}\left(9x^{2}\right)^{\frac{1}{3}}\) - step4: Rewrite the expression: \(18\times 6^{\frac{1}{3}}x^{\frac{1}{3}}\times 9^{\frac{1}{3}}x^{\frac{2}{3}}\) - step5: Multiply the numbers: \(18\times 6^{\frac{1}{3}}\times 9^{\frac{1}{3}}x^{\frac{1}{3}}\times x^{\frac{2}{3}}\) - step6: Multiply the terms: \(18\times 6^{\frac{1}{3}}\times 9^{\frac{1}{3}}x\) - step7: Simplify: \(18\sqrt[3]{6}\times \sqrt[3]{9}\times x\) - step8: Expand the expression: \(54\sqrt[3]{2}\times x\) Calculate or simplify the expression \( \sqrt((3 * m * n^2)/(5 * c^4)) * \sqrt((25 * m^6 * n)/(12 * c^(-4))) \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{\frac{\left(3mn^{2}\right)}{\left(5c^{4}\right)}}\times \sqrt{\frac{\left(25m^{6}n\right)}{\left(12c^{-4}\right)}}\) - step1: Evaluate: \(\sqrt{\frac{3mn^{2}}{5c^{4}}}\times \sqrt{\frac{25m^{6}n}{\left(12c^{-4}\right)}}\) - step2: Remove the parentheses: \(\sqrt{\frac{3mn^{2}}{5c^{4}}}\times \sqrt{\frac{25m^{6}n}{12c^{-4}}}\) - step3: Rewrite the expression: \(\sqrt{\frac{3mn^{2}}{5c^{4}}}\times \sqrt{\frac{25m^{6}nc^{4}}{12}}\) - step4: Simplify the root: \(\frac{n\sqrt{3m}}{\sqrt{5}\times c^{2}}\times \sqrt{\frac{25m^{6}nc^{4}}{12}}\) - step5: Simplify the root: \(\frac{n\sqrt{3m}}{\sqrt{5}\times c^{2}}\times \frac{5c^{2}m^{3}\sqrt{n}}{2\sqrt{3}}\) - step6: Reduce the fraction: \(\frac{n\sqrt{3m}}{\sqrt{5}}\times \frac{5m^{3}\sqrt{n}}{2\sqrt{3}}\) - step7: Multiply the terms: \(\frac{n\sqrt{3m}\times 5m^{3}\sqrt{n}}{\sqrt{5}\times 2\sqrt{3}}\) - step8: Multiply the terms: \(\frac{5nm^{3}\sqrt{3mn}}{\sqrt{5}\times 2\sqrt{3}}\) - step9: Multiply the terms: \(\frac{5nm^{3}\sqrt{3mn}}{2\sqrt{15}}\) - step10: Simplify: \(\frac{nm^{3}\sqrt{5nm}}{2}\) Calculate or simplify the expression \( 3 * (3 * a)^(1/3) * (4 * (9 * a^2)^(1/3) + 5 * (7 * a^2)^(1/3)) \). Simplify the expression by following steps: - step0: Solution: \(3\left(3a\right)^{\frac{1}{3}}\left(4\left(9a^{2}\right)^{\frac{1}{3}}+5\left(7a^{2}\right)^{\frac{1}{3}}\right)\) - step1: Multiply the terms: \(3\left(3a\right)^{\frac{1}{3}}\left(4\times 9^{\frac{1}{3}}a^{\frac{2}{3}}+5\left(7a^{2}\right)^{\frac{1}{3}}\right)\) - step2: Multiply the terms: \(3\left(3a\right)^{\frac{1}{3}}\left(4\times 9^{\frac{1}{3}}a^{\frac{2}{3}}+5\times 7^{\frac{1}{3}}a^{\frac{2}{3}}\right)\) - step3: Multiply the terms: \(3^{\frac{4}{3}}a^{\frac{1}{3}}\left(4\times 9^{\frac{1}{3}}a^{\frac{2}{3}}+5\times 7^{\frac{1}{3}}a^{\frac{2}{3}}\right)\) - step4: Simplify: \(3\sqrt[3]{3}\times a^{\frac{1}{3}}\left(4\times 9^{\frac{1}{3}}a^{\frac{2}{3}}+5\times 7^{\frac{1}{3}}a^{\frac{2}{3}}\right)\) - step5: Simplify: \(3\sqrt[3]{3}\times \sqrt[3]{a}\times \left(4\times 9^{\frac{1}{3}}a^{\frac{2}{3}}+5\times 7^{\frac{1}{3}}a^{\frac{2}{3}}\right)\) - step6: Simplify: \(3\sqrt[3]{3}\times \sqrt[3]{a}\times \left(4\sqrt[3]{9a^{2}}+5\sqrt[3]{7a^{2}}\right)\) - step7: Multiply the terms: \(3\sqrt[3]{3a}\times \left(4\sqrt[3]{9a^{2}}+5\sqrt[3]{7a^{2}}\right)\) - step8: Use the distributive property: \(3\sqrt[3]{3a}\times 4\sqrt[3]{9a^{2}}+3\sqrt[3]{3a}\times 5\sqrt[3]{7a^{2}}\) - step9: Multiply the terms: \(36a+3\sqrt[3]{3a}\times 5\sqrt[3]{7a^{2}}\) - step10: Multiply the terms: \(36a+15\sqrt[3]{21}\times a\) Calculate or simplify the expression \( (\sqrt(x) + 2 * \sqrt(y)) * (2 * \sqrt(x) - \sqrt(y)) \). Simplify the expression by following steps: - step0: Solution: \(\left(\sqrt{x}+2\sqrt{y}\right)\left(2\sqrt{x}-\sqrt{y}\right)\) - step1: Use the distributive property: \(\sqrt{x}\times 2\sqrt{x}+\sqrt{x}\times \left(-\sqrt{y}\right)+2\sqrt{y}\times 2\sqrt{x}+2\sqrt{y}\times \left(-\sqrt{y}\right)\) - step2: Calculate: \(2x+\sqrt{x}\times \left(-\sqrt{y}\right)+2\sqrt{y}\times 2\sqrt{x}+2\sqrt{y}\times \left(-\sqrt{y}\right)\) - step3: Multiply the terms: \(2x-\sqrt{xy}+2\sqrt{y}\times 2\sqrt{x}+2\sqrt{y}\times \left(-\sqrt{y}\right)\) - step4: Multiply the terms: \(2x-\sqrt{xy}+4\sqrt{yx}+2\sqrt{y}\times \left(-\sqrt{y}\right)\) - step5: Multiply the terms: \(2x-\sqrt{xy}+4\sqrt{yx}-2y\) Aquí están las soluciones a los problemas de multiplicación y simplificación de radicales: 18. \( (5 \sqrt{2})(3 \sqrt{12}) \) \[ 30 \sqrt{6} \] 19. \( (-5 \sqrt{8})(-6 \sqrt{7}) \) \[ 60 \sqrt{14} \] 20. \( \sqrt{5 a b^{7}} \sqrt{10 a^{3} b^{3}} \) \[ 5 \sqrt{2} \times b^{5} a^{2} \] 21. \( (9 \sqrt[3]{6 x})(2 \sqrt[3]{9 x^{2}}) \) \[ 54 \sqrt[3]{2} \times x \] 22. \( \sqrt{\frac{3}{4} x^{2} y^{3}} \sqrt{\frac{5}{3} x^{5} y^{-2}} \) \[ \frac{x^{3} \sqrt{5xy}}{2} \] 23. \( \sqrt{\frac{3 m n^{2}}{5 c^{4}}} \sqrt{\frac{25 m^{6} n}{12 c^{-4}}} \) \[ \frac{nm^{3} \sqrt{5nm}}{2} \] 24. \( 2 \sqrt{2 a}(3 \sqrt{12 a}-\sqrt{27 a}) \) \[ 6 \sqrt{6} \times a \] 25. \( 3 \sqrt[3]{3 a}(4 \sqrt[3]{9 a^{2}}+5 \sqrt[3]{7 a^{2}}) \) \[ 36a + 15 \sqrt[3]{21} \times a \] 26. \( (\sqrt{x}+2 \sqrt{y})(2 \sqrt{x}-\sqrt{y}) \) \[ 2x - \sqrt{xy} + 4\sqrt{yx} - 2y \] 27. \( (2 \sqrt{x}-3 \sqrt{y})(3 \sqrt{x}+5 \sqrt{y}) \) \[ 6x + 10\sqrt{xy} - 9\sqrt{yx} - 15y \] Si necesitas más ayuda o aclaraciones sobre algún paso, no dudes en preguntar.