Ejercicio 232 – Solucionario Baldor

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CAPITULO XXXI

R a d i c a l e s
Simplificación de radicales

Ejercicio 232

Simplificar

$\sqrt{\frac{1}{5}} = \sqrt{\frac{1.5}{5.5}} = \sqrt{\frac{5}{5^{2}}} = \frac{1}{5} \sqrt{5}$
$\sqrt{\frac{3}{8}} = \sqrt{\frac{3.2}{8.2}} = \sqrt{\frac{6}{2^{4}}} = \frac{1}{2^{2}} \sqrt{6} = \frac{1}{4} \sqrt{6}$
$2 \sqrt{\frac{1}{2}} = 2 \sqrt{\frac{1.2}{2.2}} = 2 \sqrt{\frac{2}{2^{2}}} = \frac{2}{2} \sqrt{2} = \sqrt{2}$
$3 \sqrt{\frac{1}{6}} = 3 \sqrt{\frac{1.6}{6.6}} = 3 \sqrt{\frac{6}{6^{2}}} = \frac{3}{} \sqrt{6} = \frac{1}{2} \sqrt{6}$
$\frac{1}{2} \sqrt{\frac{2}{3}} = \frac{1}{2} \sqrt{\frac{2.3}{3.3}} = \frac{1}{2} \sqrt{\frac{6}{3^{2}}} = \frac{1}{2.3} \sqrt{6} = \frac{1}{6} \sqrt{6}$
$\sqrt{\frac{a^{2}}{8 x}} = \sqrt{\frac{a^{2} .2 x}{8 x .2 x}} = \sqrt{\frac{2 a^{2} x}{4^{2} x^{2}}} = \frac{a}{4 x} \sqrt{2 x}$
$\begin{matrix} \frac{3}{2} \sqrt{\frac{4 a^{2}}{27 y^{3}}} & = \frac{3}{2} \sqrt{\frac{2^{2} a^{2} .3 y}{3^{3} y^{3} .3 y}} \\ = \frac{3}{2} 2 a \sqrt{\frac{3 y}{3^{4} y^{4}}} \\ = \frac{3 a}{3^{2} y^{2}} \sqrt{3 y} \\ = \frac{a}{3 y^{2}} \sqrt{3 y} \end{matrix}$
$\begin{matrix} 5 \sqrt{\frac{9 n}{5 m^{3}}} & = 5 \sqrt{\frac{9 n .5 m}{5 m^{3} .5 m}} \\ = 5 \sqrt{\frac{3^{2} n .5 m}{5^{2} m^{4}}} \\ = 5 . \frac{3}{5 m^{2}} \sqrt{5 m n} \\ = \frac{3}{m^{2}} \sqrt{5 m n} \end{matrix}$
$\begin{matrix} 6 \sqrt{\frac{5 a^{3}}{24 x^{2}}} & = 6 \sqrt{\frac{5 a^{3}}{8.3 x^{2}}} \\ = 6 \sqrt{\frac{5 a^{2} a .6}{2^{3} .3 x^{2} .6}} \\ = 6 \sqrt{\frac{5 a^{2} a .6}{2^{3} .3 x^{2} .2.3}} \\ = 6 \sqrt{\frac{30 a^{2} a}{2^{4} . 3^{2} x^{2}}} \\ = \frac{6 a}{2^{2} . 3 x} \sqrt{30 a} \\ = \frac{a}{2 x} \sqrt{30 a} \end{matrix}$
$\begin{matrix} \sqrt[3]{\frac{2}{3}} & = \sqrt[3]{\frac{2. 3^{2}}{3. 3^{2}}} \\ = \sqrt[3]{\frac{2. 3^{2}}{3^{3}}} \\ = \frac{1}{3} \sqrt[3]{18} \end{matrix}$
$\begin{matrix} 5 \sqrt[3]{\frac{1}{5}} & = 5 \sqrt[3]{\frac{1. 5^{2}}{5. 5^{2}}} \\ = 5 \sqrt[3]{\frac{5^{2}}{5^{3}}} \\ = \frac{5}{5} \sqrt[3]{25} \\ = \sqrt[3]{25} \end{matrix}$
$\begin{matrix} \sqrt[3]{\frac{8}{9 x^{2}}} & = \sqrt[3]{\frac{8.3 x}{3^{2} x^{2} .3 x}} \\ = \sqrt[3]{\frac{24 x}{3^{3} x^{3}}} \\ = \frac{1}{3 x} \sqrt[3]{24 x} \end{matrix}$
$\begin{matrix} 2 b^{2} \sqrt[3]{\frac{125}{4 b^{5}}} & = 2 b^{2} \sqrt[3]{\frac{5^{3}}{2^{2} . b^{5}}} \\ = 2 b^{2} \sqrt[3]{\frac{5^{3} .2 b}{2^{2} . b^{5} .2 b}} \\ = 2 b^{2} \sqrt[3]{\frac{5^{3} .2 b}{2^{3} . b^{6}}} \\ = \frac{2 b^{2}}{2 b^{2}} 5 \sqrt[3]{2 b} \\ = 5 \sqrt[3]{2 b} \end{matrix}$
$\begin{matrix} \frac{2}{3} \sqrt[3]{\frac{27 x^{2}}{16 a^{2} b^{4}}} & = \frac{2}{3} \sqrt[3]{\frac{3^{3} . x^{2}}{4^{2} a^{2} b^{4}}} \\ = \frac{2}{3} \sqrt[3]{\frac{3^{3} . x^{2} .4 a b^{2}}{4^{2} a^{2} b^{4} .4 a b^{2}}} \\ = \frac{2}{3} \sqrt[3]{\frac{3^{3} . x^{2} .4 a b^{2}}{4^{3} a^{3} b^{6}}} \\ = \frac{2}{3 . a b^{2}} 3 \sqrt[3]{4 a b^{2} x^{2}} \\ = \frac{1}{2 a b^{2}} \sqrt[3]{4 a b^{2} x^{2}} \end{matrix}$
$\begin{matrix} 2 x y \sqrt[4]{\frac{81 a^{2}}{4 x^{3} y}} & = 2 x y \sqrt[4]{\frac{3^{4} a^{2}}{2^{2} x^{3} y}} \\ = 2 x y \sqrt[4]{\frac{3^{4} a^{2} . 2^{2} x y^{3}}{2^{2} x^{3} y . 2^{2} x y^{3}}} \\ = 2 x y \sqrt[4]{\frac{3^{4} a^{2} . 2^{2} x y^{3}}{2^{4} x^{4} y^{4}}} \\ = \frac{2 x y}{2 x y} 3 \sqrt[4]{4 a^{2} x y^{3}} \\ = 3 \sqrt[4]{4 a^{2} x y^{3}} \end{matrix}$