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4) Simplifique al \( \sqrt[3]{128}-\sqrt[3]{250}+\sqrt[3]{54}-\sqrt[3]{16} \) b) \( \sqrt[3]{375}-\sqrt[3]{24}+\sqrt[3]{81}-\sqrt[3]{192} \) c) \( a \sqrt[3]{a b^{4}}+b \sqrt[3]{a^{4} b}+\sqrt[3]{0^{4} b^{4}}-3 a b \sqrt[3]{a b} \)

Ask by Harmon O'Quinn. in Brazil
Mar 12,2025

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a) \( \sqrt[3]{128} - \sqrt[3]{250} + \sqrt[3]{54} - \sqrt[3]{16} = 0 \) b) \( \sqrt[3]{375} - \sqrt[3]{24} + \sqrt[3]{81} - \sqrt[3]{192} = 2\sqrt[3]{3} \) c) \( a \sqrt[3]{a b^{4}} + b \sqrt[3]{a^{4} b} + \sqrt[3]{0^{4} b^{4}} - 3 a b \sqrt[3]{a b} = 2\sqrt[3]{ab} \times ab - 3ab\sqrt[3]{ab} \)

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Para simplificar la expresión \( \sqrt[3]{128}-\sqrt[3]{250}+\sqrt[3]{54}-\sqrt[3]{16} \), primero descomponemos los números en factores primos. \( 128 = 2^7 \) así que \( \sqrt[3]{128} = 2^{7/3} = 2^2 \cdot 2^{1/3} = 4\sqrt[3]{2} \). Para \( 250 = 2 \times 5^3 \), tenemos \( \sqrt[3]{250} = 5 \sqrt[3]{2} \). Luego \( 54 = 2 \cdot 3^3 \), así que \( \sqrt[3]{54} = 3 \sqrt[3]{2} \). Finalmente, para \( 16 = 2^4 \), se obtiene \( \sqrt[3]{16} = 2^{4/3} = 2^{1} \cdot 2^{1/3} = 2\sqrt[3]{2} \). Al juntar todo esto: \[ 4\sqrt[3]{2} - 5\sqrt[3]{2} + 3\sqrt[3]{2} - 2\sqrt[3]{2} = (4 - 5 + 3 - 2)\sqrt[3]{2} = 0\sqrt[3]{2} = 0 \] Para \( \sqrt[3]{375}-\sqrt[3]{24}+\sqrt[3]{81}-\sqrt[3]{192} \), el número \( 375 = 3 \times 5^3 \), así que \( \sqrt[3]{375} = 5 \sqrt[3]{3} \). Para \( 24 = 2^3 \cdot 3 \), se obtiene \( \sqrt[3]{24} = 2\sqrt[3]{3} \). En cuanto a \( 81 = 3^4 \), así que \( \sqrt[3]{81} = 3\sqrt[3]{3} \); y para \( 192 = 2^6 \cdot 3 \), obtenemos \( \sqrt[3]{192} = 4\sqrt[3]{3} \). Entonces, uniendo todo: \[ 5\sqrt[3]{3} - 2\sqrt[3]{3} + 3\sqrt[3]{3} - 4\sqrt[3]{3} = (5 - 2 + 3 - 4)\sqrt[3]{3} = 2\sqrt[3]{3} \] Finalmente, para \( a \sqrt[3]{a b^{4}} + b \sqrt[3]{a^{4} b} + \sqrt[3]{0^{4} b^{4}} - 3 a b \sqrt[3]{a b} \) notamos que, dado que \( \sqrt[3]{0^{4} b^{4}} = 0 \), simplificamos a: \[ a \sqrt[3]{a b^{4}} + b \sqrt[3]{a^{4} b} - 3ab \sqrt[3]{ab} \] Ahora, utilizando la propiedad de la raíz cúbica y combinando términos similares, obtenemos: \[ \sqrt[3]{a^2b^4} + \sqrt[3]{a^4b} - 3ab \sqrt[3]{ab} \] La propiedad y simplificaciones adicionales permitirán expresar esto en un formato más compacto dependiendo de \( a \) y \( b \). ¡Matemáticas al rescate!

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