Ejercicio 247

octubre 30, 2019abril 14, 2019 por Gestor Baldor

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CAPITULO XXXI

R a d i c a l e s
Racionalización

Ejercicio 247

Racionalizar el denominador de:

$\begin{matrix} \frac{1}{\sqrt{3}} & = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \\ = \frac{\sqrt{3}}{\sqrt{3^{2}}} \\ = \frac{\sqrt{3}}{3} \end{matrix}$
$\begin{matrix} \frac{5}{\sqrt{2}} & = \frac{5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} \\ = \frac{5 \sqrt{2}}{\sqrt{2^{2}}} \\ = \frac{5 \sqrt{2}}{2} \end{matrix}$
$\begin{matrix} \frac{3}{4 \sqrt{5}} & = \frac{3}{4 \sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \\ = \frac{3 \sqrt{5}}{4 \sqrt{5^{2}}} \\ = \frac{3 \sqrt{5}}{4 (5)} \\ = \frac{3 \sqrt{5}}{20} \end{matrix}$
$\begin{matrix} \frac{2 a}{\sqrt{2 a x}} & = \frac{2 a}{\sqrt{2 a x}} \times \frac{\sqrt{2 a x}}{\sqrt{2 a x}} \\ = \frac{2 a \sqrt{2 a x}}{\sqrt{{(2 a x)}^{2}}} \\ = \frac{2 a \sqrt{2 a x}}{2 a x} \\ = \frac{\sqrt{2 a x}}{x} \end{matrix}$
$\begin{matrix} \frac{5}{\sqrt[3]{4 a^{2}}} & = \frac{5}{\sqrt[3]{2^{2} a^{2}}} \times \frac{\sqrt[3]{2 a}}{\sqrt[3]{2 a}} \\ = \frac{5 \sqrt[3]{2 a}}{\sqrt[3]{{(2 a)}^{3}}} \\ = \frac{5 \sqrt[3]{2 a}}{2 a} \end{matrix}$
$\begin{matrix} \frac{1}{\sqrt[3]{9 x}} & = \frac{1}{\sqrt[3]{3^{2} x}} \times \frac{\sqrt[3]{3 x^{2}}}{\sqrt[3]{3 x^{2}}} \\ = \frac{\sqrt[3]{3 x^{2}}}{\sqrt[3]{{(3 x)}^{3}}} \\ = \frac{\sqrt[3]{3 x^{2}}}{3 x} \end{matrix}$
$\begin{matrix} \frac{3}{\sqrt[4]{9 a}} & = \frac{3}{\sqrt[4]{3^{2} a}} \times \frac{\sqrt[4]{3^{2} a^{3}}}{\sqrt[4]{3^{2} a^{3}}} \\ = \frac{3 \sqrt[4]{3^{2} a^{3}}}{\sqrt[4]{{(3 a)}^{4}}} \\ = \frac{3 \sqrt[4]{9 a^{3}}}{3 a} \\ = \frac{\sqrt[4]{9 a^{3}}}{a} \end{matrix}$
$\begin{matrix} \frac{6}{5 \sqrt[3]{3 x}} & = \frac{6}{5 \sqrt[3]{3 x}} \times \frac{\sqrt[3]{{(3 x)}^{2}}}{\sqrt[3]{{(3 x)}^{2}}} \\ = \frac{6 \sqrt[3]{{(3 x)}^{2}}}{5 \sqrt[3]{{(3 x)}^{3}}} \\ = \frac{\sqrt[3]{9 x^{2}}}{5 (3 x)} \\ = \frac{2 \sqrt[3]{9 x^{2}}}{5 x} \end{matrix}$
$\begin{matrix} \frac{x}{\sqrt[4]{27 x^{2}}} & = \frac{x}{\sqrt[4]{3^{3} x^{2}}} \times \frac{\sqrt[4]{3 x^{2}}}{\sqrt[4]{3 x^{2}}} \\ = \frac{x \sqrt[4]{3 x^{2}}}{\sqrt[4]{{(3 x)}^{4}}} \\ = \frac{x \sqrt[4]{3 x^{2}}}{3 x} \\ = \frac{\sqrt[4]{3 x^{2}}}{3} \end{matrix}$
$\begin{matrix} \frac{1}{\sqrt[5]{8 a^{4}}} & = \frac{1}{\sqrt[5]{2^{3} a^{4}}} \times \frac{\sqrt[5]{2^{2} a}}{\sqrt[5]{2^{2} a}} \\ = \frac{\sqrt[5]{2^{2} a}}{\sqrt[5]{{(2 a)}^{5}}} \\ = \frac{\sqrt[5]{4 a}}{2 a} \end{matrix}$
$\begin{matrix} \frac{5 n^{2}}{3 \sqrt{m n}} & = \frac{5 n^{2}}{3 \sqrt{m n}} \times \frac{\sqrt{m n}}{\sqrt{m n}} \\ = \frac{5 n^{2} \sqrt{m n}}{3 \sqrt{{(m n)}^{2}}} \\ = \frac{5 n^{2} \sqrt{m n}}{3 m n} \\ = \frac{5 n \sqrt{m n}}{3 m} \end{matrix}$
$\begin{matrix} \frac{1}{5 a \sqrt[4]{25 x^{3}}} & = \frac{1}{5 a \sqrt[4]{5^{2} x^{3}}} \times \frac{\sqrt[4]{5^{2} x}}{\sqrt[4]{5^{2} x}} \\ = \frac{\sqrt[4]{5^{2} x}}{5 a \sqrt[4]{{(5 x)}^{4}}} \\ = \frac{\sqrt[4]{25 x}}{5 a (5 x)} \\ = \frac{\sqrt[4]{25 x}}{25 a x} \end{matrix}$