Fowler Donnelly
05/31/2023 · Primary School
(Des. Alto) Al resolver la expresión \( \sqrt[4]{\frac{4.096}{16}} \) aplicando las propiedades de la radicación el resultado es \( \begin{array}{llll}\text { A. } 4 & \text { B. } 9 & \text { C. } 2 & \text { D. } 12\end{array} \)
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Step-by-step Solution
Para resolver la expresión \( \sqrt[4]{\frac{4.096}{16}} \), primero simplificamos la fracción dentro de la raíz.
Calculamos \( \frac{4.096}{16} \):
\[
4.096 \div 16 = 0.256
\]
Ahora, tenemos que calcular \( \sqrt[4]{0.256} \).
Podemos expresar \( 0.256 \) como una potencia de \( 2 \):
\[
0.256 = \frac{256}{1000} = \frac{256}{100} = \frac{256}{10^2} = \frac{2^8}{10^2} = \frac{2^8}{(2 \cdot 5)^2} = \frac{2^8}{2^2 \cdot 5^2} = \frac{2^{8-2}}{5^2} = \frac{2^6}{5^2}
\]
Sin embargo, es más sencillo calcular \( \sqrt[4]{0.256} \) directamente.
Sabemos que:
\[
0.256 = \left(0.4\right)^2
\]
Por lo tanto:
\[
\sqrt[4]{0.256} = \sqrt[4]{(0.4)^2} = \sqrt{0.4}
\]
Ahora, calculamos \( \sqrt{0.4} \):
\[
\sqrt{0.4} = \sqrt{\frac{4}{10}} = \sqrt{\frac{2^2}{10}} = \frac{2}{\sqrt{10}} = \frac{2\sqrt{10}}{10} = \frac{\sqrt{10}}{5}
\]
Sin embargo, para simplificar el cálculo, podemos volver a la fracción original:
\[
\sqrt[4]{\frac{4.096}{16}} = \frac{\sqrt[4]{4.096}}{\sqrt[4]{16}} = \frac{\sqrt[4]{4.096}}{2}
\]
Calculamos \( \sqrt[4]{4.096} \):
Sabemos que \( 4.096 = 2^{12} \), por lo que:
\[
\sqrt[4]{4.096} = \sqrt[4]{2^{12}} = 2^{12/4} = 2^3 = 8
\]
Entonces:
\[
\sqrt[4]{\frac{4.096}{16}} = \frac{8}{2} = 4
\]
Por lo tanto, el resultado es:
\[
\boxed{4}
\]
Quick Answer
El resultado es 4.
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