Ejercicio 244 – Solucionario Baldor

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DESCARGA

CAPITULO XXXI

R a d i c a l e s
División de radicales de distinto índice

Ejercicio 244

Dividir:

$\begin{matrix} \sqrt[3]{2} \div \sqrt{2} \\ \sqrt[3]{2} = \sqrt[6]{2^{2}} \\ \sqrt{2} = \sqrt[6]{2^{3}} \\ \sqrt[3]{2} \div \sqrt{2} & = \sqrt[6]{2^{2}} \div \sqrt[6]{2^{3}} \\ = \sqrt[6]{\frac{2^{2}}{2^{3}}} \\ = \sqrt[6]{\frac{1}{2}} \\ = \sqrt[6]{\frac{1}{2} \times \frac{2^{5}}{2^{5}}} \\ = \sqrt[6]{\frac{2^{5}}{2^{6}}} \\ = \frac{1}{2} \sqrt[6]{2^{5}} \\ = \frac{1}{2} \sqrt[6]{32} \end{matrix}$
$\begin{matrix} \sqrt{9 x} \div \sqrt[3]{3 x^{2}} \\ \sqrt{9 x} = \sqrt{3^{2} x} = 3 \sqrt{x} = 3 \sqrt[6]{x^{3}} \\ \sqrt[3]{3 x^{2}} = \sqrt[6]{{(3 x^{2})}^{2}} = \sqrt[6]{3^{2} x^{4}} \\ \sqrt{9 x} \div \sqrt[3]{3 x^{2}} & = 3 \sqrt[6]{x^{3}} \div \sqrt[6]{3^{2} x^{4}} \\ = 3 \sqrt[6]{\frac{x^{3}}{3^{2} x^{4}}} \\ = 3 \sqrt[6]{\frac{1}{3^{2} x}} \\ = 3 \sqrt[6]{\frac{1}{3^{2} x} \times \frac{3^{4} x^{5}}{3^{4} x^{5}}} \\ = 3 \sqrt[6]{\frac{3^{4} x^{5}}{3^{6} x^{6}}} \\ = \frac{3}{3 x} \sqrt[6]{3^{4} x^{5}} \\ = \frac{1}{x} \sqrt[6]{81 x^{5}} \end{matrix}$
$\begin{matrix} \sqrt[3]{8 a^{3} b} \div \sqrt[4]{4 a^{2}} \\ \sqrt[3]{8 a^{3} b} = \sqrt[3]{2^{3} a^{3} b} = 2 a \sqrt[3]{b} = 2 a \sqrt[12]{b^{4}} \\ \sqrt[4]{4 a^{2}} = \sqrt[12]{{(2^{2} a^{2})}^{3}} = \sqrt[12]{2^{6} a^{6}} \\ \sqrt[3]{8 a^{3} b} \div \sqrt[4]{4 a^{2}} & = 2 a \sqrt[12]{b^{4}} \div \sqrt[12]{2^{6} a^{6}} \\ = 2 a \sqrt[12]{\frac{b^{4}}{2^{6} a^{6}}} \\ = 2 a \sqrt[12]{\frac{b^{4}}{2^{6} a^{6}} \times \frac{2^{6} a^{6}}{2^{6} a^{6}}} \\ = 2 a \sqrt[12]{\frac{2^{6} a^{6} b^{4}}{2^{12} a^{12}}} \\ = \frac{2 a}{2 a} \sqrt[12]{{(2 a)}^{6} b^{4}} \\ = \sqrt{2 a} \sqrt[3]{b} \end{matrix}$
$\begin{matrix} \frac{1}{2} \sqrt{2 x} \div \frac{1}{4} \sqrt[6]{16 x^{4}} \\ \frac{1}{2} \sqrt{2 x} = \frac{1}{2} \sqrt[6]{{(2 x)}^{3}} \\ \frac{1}{4} \sqrt[6]{16 x^{4}} = \frac{1}{4} \sqrt[6]{{(2 x)}^{4}} \\ \frac{1}{2} \sqrt{2 x} \div \frac{1}{4} \sqrt[6]{16 x^{4}} & = \frac{1}{2} \sqrt[6]{{(2 x)}^{3}} \div \frac{1}{4} \sqrt[6]{{(2 x)}^{4}} \\ = \frac{\frac{1}{2} \sqrt[6]{{(2 x)}^{3}}}{\frac{1}{} \sqrt[6]{{(2 x)}^{4}}} \\ = 2 \sqrt[6]{\frac{{(2 x)}^{3}}{{(2 x)}^{4}}} \\ = 2 \sqrt[6]{\frac{1}{2 x} \times \frac{{(2 x)}^{5}}{{(2 x)}^{5}}} \\ = 2 \sqrt[6]{\frac{{(2 x)}^{5}}{{(2 x)}^{6}}} \\ = 2 \frac{1}{2 x} \sqrt[6]{{(2 x)}^{5}} \\ = \frac{1}{x} \sqrt[6]{32 x^{5}} \end{matrix}$
$\begin{matrix} \sqrt[3]{5 m^{2} n} \div \sqrt[5]{m^{3} n^{2}} \\ \sqrt[3]{5 m^{2} n} = \sqrt[15]{{(5 m^{2} n)}^{5}} = \sqrt[15]{5^{5} m^{10} n^{5}} \\ \sqrt[5]{m^{3} n^{2}} = \sqrt[15]{{(m^{3} n^{2})}^{3}} = \sqrt[15]{m^{9} n^{6}} \\ \sqrt[3]{5 m^{2} n} \div \sqrt[5]{m^{3} n^{2}} & = \sqrt[15]{5^{5} m^{10} n^{5}} \div \sqrt[15]{m^{9} n^{6}} \\ = \sqrt[15]{\frac{5^{5} m^{10} n^{5}}{m^{9} n^{6}}} \\ = \sqrt[15]{\frac{5^{5} m}{n} \times \frac{n^{14}}{n^{14}}} \\ = \sqrt[15]{\frac{5^{5} m n^{14}}{n^{15}}} \\ = \frac{1}{n} \sqrt[15]{3125 m n^{14}} \end{matrix}$
$\begin{matrix} \sqrt[6]{18 x^{3} y^{4} z^{5}} \div \sqrt[4]{3 x^{2} y^{2} z^{3}} \\ \sqrt[6]{18 x^{3} y^{4} z^{5}} = \sqrt[12]{{(18 x^{3} y^{4} z^{5})}^{2}} = \sqrt[12]{1 8^{2} x^{6} y^{8} z^{10}} \\ \sqrt[4]{3 x^{2} y^{2} z^{3}} = \sqrt[12]{{(3 x^{2} y^{2} z^{3})}^{3}} = \sqrt[12]{3^{3} x^{6} y^{6} z^{9}} \\ \sqrt[6]{18 x^{3} y^{4} z^{5}} \div \sqrt[4]{3 x^{2} y^{2} z^{3}} & = \sqrt[12]{1 8^{2} x^{6} y^{8} z^{10}} \div \sqrt[12]{3^{3} x^{6} y^{6} z^{9}} \\ = \sqrt[12]{\frac{2^{2} . 3^{4} x^{6} y^{} z^{10}}{3^{3} x^{6} y^{6} z^{9}}} \\ = \sqrt[12]{12 y^{2} z} \end{matrix}$
$\begin{matrix} \sqrt[3]{3 m^{4}} \div \sqrt[9]{27 m^{2}} \\ \sqrt[3]{3 m^{4}} = \sqrt[9]{{(3 m^{4})}^{3}} = \sqrt[9]{3^{3} m^{12}} \\ \sqrt[9]{27 m^{2}} = \sqrt[9]{3^{3} m^{2}} \\ \sqrt[3]{3 m^{4}} \div \sqrt[9]{27 m^{2}} & = \sqrt[9]{3^{3} m^{12}} \div \sqrt[9]{3^{3} m^{2}} \\ = \sqrt[9]{\frac{3^{3} m^{}}{3^{3} m^{2}}} \\ = \sqrt[9]{m^{10}} \\ = \sqrt[9]{m^{9} . m} \\ = m \sqrt[9]{m} \end{matrix}$
$\begin{matrix} \frac{4}{5} \sqrt[3]{4 a b} \div \frac{1}{10} \sqrt{2 a^{2}} \\ \frac{4}{5} \sqrt[3]{4 a b} = \frac{4}{5} \sqrt[6]{{(4 a b)}^{2}} = \frac{4}{5} \sqrt[6]{2^{4} a^{2} b^{2}} \\ \frac{1}{10} \sqrt{2 a^{2}} = \frac{1}{10} \sqrt[6]{{(2 a^{2})}^{3}} = \frac{1}{10} \sqrt[6]{2^{3} a^{6}} \\ \frac{4}{5} \sqrt[3]{4 a b} \div \frac{1}{10} \sqrt{2 a^{2}} & = \frac{4}{5} \sqrt[6]{2^{4} a^{2} b^{2}} \div \frac{1}{10} \sqrt[6]{2^{3} a^{6}} \\ = \frac{\frac{4}{5} \sqrt[6]{2^{4} a^{2} b^{2}}}{\frac{1}{} \sqrt[6]{2^{3} a^{6}}} \\ = 8 \sqrt[6]{\frac{2^{4} a^{2} b^{2}}{2^{3} a^{}}} \\ = 8 \sqrt[6]{\frac{2 b^{2}}{a^{4}} \times \frac{a^{2}}{a^{2}}} \\ = 8 \sqrt[6]{\frac{2 a^{2} b^{2}}{a^{6}}} \\ = \frac{8}{a} \sqrt[6]{2 a^{2} b^{2}} \end{matrix}$