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2) Resuelve utiliz a) \( \frac{\left(3^{-2}\right)\left(3^{6}\right)(2}{(9)\left(5^{-6}\right)} \) b) \( \frac{0.5^{\frac{5}{2}} \cdot\left(\frac{1}{2}\right)^{\frac{1}{4}}}{0.5^{\frac{1}{2}}} \)

Ask by Guerrero Boyd. in Colombia
Mar 15,2025

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**a) Simplificación de** \[ \frac{\left(3^{-2}\right)\left(3^{6}\right)\cdot 2}{9\cdot\left(5^{-6}\right)} = 18 \cdot 5^6 \quad \text{ó} \quad 281250 \] **b) Simplificación de** \[ \frac{0.5^{\frac{5}{2}} \cdot\left(\frac{1}{2}\right)^{\frac{1}{4}}}{0.5^{\frac{1}{2}}} = \left(\frac{1}{2}\right)^{\frac{9}{4}} \quad \text{ó} \quad 2^{-\frac{9}{4}} \]

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The Deep Dive

Para resolver la primera expresión \( \frac{\left(3^{-2}\right)\left(3^{6}\right)(2}{(9)\left(5^{-6}\right)} \): Primero, simplificamos dentro del numerador: \( 3^{-2} \cdot 3^{6} = 3^{(-2 + 6)} = 3^{4} \). El 9 en el denominador se puede escribir como \( 3^{2} \), así que ahora tenemos: \[ \frac{(3^{4} \cdot 2)}{(3^{2} \cdot 5^{-6})} = \frac{3^{4} \cdot 2}{3^{2}} \cdot \frac{1}{5^{-6}} = 3^{(4 - 2)} \cdot 2 \cdot 5^{6} = 3^{2} \cdot 2 \cdot 5^{6} \] Finalmente, \( 3^{2} = 9 \), así que todo se convierte en \( 9 \cdot 2 \cdot 15625 = 18 \cdot 15625 = 281250 \). Para \( \frac{0.5^{\frac{5}{2}} \cdot\left(\frac{1}{2}\right)^{\frac{1}{4}}}{0.5^{\frac{1}{2}}} \): Empezamos simplificando el numerador: \[ 0.5^{\frac{5}{2}} = (2^{-1})^{\frac{5}{2}} = 2^{-\frac{5}{2}} \] Y \[ \left(\frac{1}{2}\right)^{\frac{1}{4}} = (2^{-1})^{\frac{1}{4}} = 2^{-\frac{1}{4}} \] Entonces, el numerador se convierte en: \[ 2^{-\frac{5}{2}} \cdot 2^{-\frac{1}{4}} = 2^{-\left(\frac{5}{2} + \frac{1}{4}\right)} \] Para convertir, necesitamos un denominador común. Así que convertimos \(\frac{5}{2} = \frac{10}{4}\): \[ -\left(\frac{10}{4} + \frac{1}{4}\right) = -\frac{11}{4} \] Ahora el numerador queda \( 2^{-\frac{11}{4}} \) y el denominador: \[ 0.5^{\frac{1}{2}} = (2^{-1})^{\frac{1}{2}} = 2^{-\frac{1}{2}} \] Así que ahora tenemos: \[ \frac{2^{-\frac{11}{4}}}{2^{-\frac{1}{2}}} = 2^{-\left(\frac{11}{4} - \frac{2}{4}\right)} = 2^{-\frac{9}{4}} \] Por lo tanto, el resultado es \( \frac{1}{2^{\frac{9}{4}}} \).

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