Carlson Spencer
03/08/2024 · High School
Simplifique al máximo cada expresión. a. \( \left(27 x^{9}\right)^{\frac{1}{3}} \) b. \( \left(3 x^{\frac{1}{2}} y^{\frac{3}{2}}\right)^{2} \) c. \( \left(\left(-4 x^{6} y^{-3}\right)\left(-2 x^{-2} y^{2}\right)\right)^{\frac{1}{2}} \) d. \( \frac{\left(x^{-\frac{1}{8}} y\right)^{-2}\left(x^{4} y^{6}\right)^{-\frac{1}{2}}}{x^{\frac{1}{4}} y^{4}} \)
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Step-by-step Solution
Vamos a simplificar cada una de las expresiones paso a paso.
a. \( \left(27 x^{9}\right)^{\frac{1}{3}} \)
Primero, aplicamos la propiedad de las potencias:
\[
\left(27\right)^{\frac{1}{3}} \cdot \left(x^{9}\right)^{\frac{1}{3}} = 3 \cdot x^{3} = 3x^{3}
\]
b. \( \left(3 x^{\frac{1}{2}} y^{\frac{3}{2}}\right)^{2} \)
Aplicamos la propiedad de las potencias:
\[
3^{2} \cdot \left(x^{\frac{1}{2}}\right)^{2} \cdot \left(y^{\frac{3}{2}}\right)^{2} = 9 \cdot x^{1} \cdot y^{3} = 9xy^{3}
\]
c. \( \left(\left(-4 x^{6} y^{-3}\right)\left(-2 x^{-2} y^{2}\right)\right)^{\frac{1}{2}} \)
Primero, multiplicamos los términos dentro del paréntesis:
\[
(-4)(-2) \cdot x^{6} \cdot x^{-2} \cdot y^{-3} \cdot y^{2} = 8 \cdot x^{6 - 2} \cdot y^{-3 + 2} = 8x^{4}y^{-1}
\]
Ahora aplicamos la raíz cuadrada:
\[
\left(8x^{4}y^{-1}\right)^{\frac{1}{2}} = \sqrt{8} \cdot \left(x^{4}\right)^{\frac{1}{2}} \cdot \left(y^{-1}\right)^{\frac{1}{2}} = 2\sqrt{2} \cdot x^{2} \cdot y^{-\frac{1}{2}} = 2\sqrt{2} x^{2} \cdot \frac{1}{\sqrt{y}} = \frac{2\sqrt{2} x^{2}}{\sqrt{y}}
\]
d. \( \frac{\left(x^{-\frac{1}{8}} y\right)^{-2}\left(x^{4} y^{6}\right)^{-\frac{1}{2}}}{x^{\frac{1}{4}} y^{4}} \)
Primero simplificamos el numerador:
\[
\left(x^{-\frac{1}{8}} y\right)^{-2} = x^{\frac{1}{4}} y^{-2}
\]
\[
\left(x^{4} y^{6}\right)^{-\frac{1}{2}} = x^{-2} y^{-3}
\]
Ahora combinamos:
\[
x^{\frac{1}{4}} y^{-2} \cdot x^{-2} y^{-3} = x^{\frac{1}{4} - 2} y^{-2 - 3} = x^{-\frac{7}{4}} y^{-5}
\]
Ahora, el numerador es \( x^{-\frac{7}{4}} y^{-5} \) y el denominador es \( x^{\frac{1}{4}} y^{4} \):
\[
\frac{x^{-\frac{7}{4}} y^{-5}}{x^{\frac{1}{4}} y^{4}} = x^{-\frac{7}{4} - \frac{1}{4}} y^{-5 - 4} = x^{-2} y^{-9}
\]
Finalmente, podemos escribirlo como:
\[
\frac{1}{x^{2} y^{9}}
\]
Resumiendo las respuestas:
a. \( 3x^{3} \)
b. \( 9xy^{3} \)
c. \( \frac{2\sqrt{2} x^{2}}{\sqrt{y}} \)
d. \( \frac{1}{x^{2} y^{9}} \)
Quick Answer
a. \( 3x^{3} \)
b. \( 9xy^{3} \)
c. \( \frac{2\sqrt{2} x^{2}}{\sqrt{y}} \)
d. \( \frac{1}{x^{2} y^{9}} \)
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