Ejercicio 242 – Solucionario Baldor

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DESCARGA

CAPITULO XXXI

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

Ejercicio 242

Multiplicar:

$\begin{matrix} \sqrt{x} \times \sqrt[3]{2 x^{2}} \\ \sqrt{x} = \sqrt[6]{x^{3}} \\ \sqrt[3]{2 x^{2}} = \sqrt[6]{{(2 x^{2})}^{2}} = \sqrt[6]{4 x^{4}} \\ \sqrt{x} \times \sqrt[3]{2 x^{2}} & = \sqrt[6]{x^{3}} \times \sqrt[6]{4 x^{4}} \\ = \sqrt[6]{4 x^{7}} \\ = \sqrt[6]{4 x^{6} x} \\ = x \sqrt[6]{4 x} \end{matrix}$
$\begin{matrix} 3 \sqrt{2 a b} \times 4 \sqrt[4]{8 a^{3}} \\ 3 \sqrt{2 a b} = 3 \sqrt[4]{{(2 a b)}^{2}} = 3 \sqrt[4]{2^{2} a^{2} b^{2}} \\ 3 \sqrt{2 a b} \times 4 \sqrt[4]{8 a^{3}} & = 3 \sqrt[4]{2^{2} a^{2} b^{2}} \times 4 \sqrt[4]{2^{3} a^{3}} \\ = 12 \sqrt[4]{2^{5} a^{5} b^{2}} \\ = 12 \times 2 a \sqrt[4]{2 a b^{2}} \\ = 24 a \sqrt[4]{2 a b^{2}} \end{matrix}$
$\begin{matrix} \sqrt[3]{9 x^{2} y} \times \sqrt[6]{81 x^{5}} \\ \sqrt[3]{9 x^{2} y} = \sqrt[6]{{(9 x^{2} y)}^{2}} = \sqrt[6]{81 x^{4} y^{2}} \\ \sqrt[3]{9 x^{2} y} \times \sqrt[6]{81 x^{5}} & = \sqrt[6]{81 x^{4} y^{2}} \times \sqrt[6]{81 x^{5}} \\ = \sqrt[6]{8 1^{2} x^{9} y^{2}} \\ = \sqrt[6]{{(3^{4})}^{2} x^{9} y^{2}} \\ = \sqrt[6]{3^{8} x^{9} y^{2}} \\ = 3 x \sqrt[6]{3^{2} x^{3} y^{2}} \end{matrix}$
$\begin{matrix} \sqrt[3]{a^{2} b^{2}} \times 2 \sqrt[4]{3 a^{3} b} \\ \sqrt[3]{a^{2} b^{2}} = \sqrt[12]{{(a^{2} b^{2})}^{4}} = \sqrt[12]{a^{8} b^{8}} \\ 2 \sqrt[4]{3 a^{3} b} = 2 \sqrt[12]{{(3 a^{3} b)}^{3}} = 2 \sqrt[12]{27 a^{9} b^{3}} \\ \sqrt[3]{a^{2} b^{2}} \times 2 \sqrt[4]{3 a^{3} b} & = \sqrt[12]{a^{8} b^{8}} \times 2 \sqrt[12]{27 a^{9} b^{3}} \\ = 2 \sqrt[12]{27 a^{17} b^{11}} \\ = 2 a \sqrt[12]{27 a^{5} b^{11}} \end{matrix}$
$\begin{matrix} \sqrt[4]{25 x^{2} y^{3}} \times \sqrt[6]{125 x^{2}} \\ \sqrt[4]{25 x^{2} y^{3}} = \sqrt[12]{{(5^{2} x^{2} y^{3})}^{3}} = \sqrt[12]{5^{6} x^{6} y^{9}} \\ \sqrt[6]{125 x^{2}} = \sqrt[12]{{(5^{3} x^{2})}^{2}} = \sqrt[12]{5^{6} x^{4}} \\ \sqrt[4]{25 x^{2} y^{3}} \times \sqrt[6]{125 x^{2}} & = \sqrt[12]{5^{6} x^{6} y^{9}} \times \sqrt[12]{5^{6} x^{4}} \\ = \sqrt[12]{5^{12} x^{10} y^{9}} \\ = 5 \sqrt[12]{x^{10} y^{9}} \end{matrix}$
$\begin{matrix} \frac{2}{3} \sqrt[3]{4 m^{2}} \times \frac{3}{4} \sqrt[5]{16 m^{4} n} \\ \frac{2}{3} \sqrt[3]{4 m^{2}} = \frac{2}{3} \sqrt[15]{{(2^{2} m^{2})}^{5}} = \frac{2}{3} \sqrt[15]{2^{10} m^{10}} \\ \frac{3}{4} \sqrt[5]{16 m^{4} n} = \frac{3}{4} \sqrt[15]{{(2^{4} m^{4} n)}^{3}} = \frac{3}{4} \sqrt[15]{2^{12} m^{12} n^{3}} \\ \frac{2}{3} \sqrt[3]{4 m^{2}} \times \frac{3}{4} \sqrt[5]{16 m^{4} n} & = \frac{2}{3} \sqrt[15]{2^{10} m^{10}} \times \frac{3}{4} \sqrt[15]{2^{12} m^{12} n^{3}} \\ = \frac{2}{3} \times \frac{3}{} \sqrt[15]{2^{22} m^{22} n^{3}} \\ = \frac{1}{2} \times 2 m \sqrt[15]{2^{7} m^{7} n^{3}} \\ = m \sqrt[15]{2^{7} m^{7} n^{3}} \end{matrix}$
$\begin{matrix} \sqrt{\frac{1}{2 x}} \times \sqrt[3]{x^{2}} \\ \sqrt{\frac{1}{2 x}} = \sqrt{\frac{2 x}{{(2 x)}^{2}}} = \frac{1}{2 x} \sqrt{2 x} = \frac{1}{2 x} \sqrt[6]{{(2 x)}^{3}} = \frac{1}{2 x} \sqrt[6]{2^{3} x^{3}} \\ \sqrt[3]{x^{2}} = \sqrt[6]{{(x^{2})}^{2}} = \sqrt[6]{x^{4}} \\ \sqrt{\frac{1}{2 x}} \times \sqrt[3]{x^{2}} & = \frac{1}{2 x} \sqrt[6]{2^{3} x^{3}} \times \sqrt[6]{x^{4}} \\ = \frac{1}{2 x} \sqrt[6]{2^{3} x^{7}} \\ = \frac{1}{2 x} x \sqrt[6]{8 x} \\ = \frac{1}{2} \sqrt[6]{8 x} \end{matrix}$
$\begin{matrix} \sqrt{2 x} \times \sqrt[5]{4 x} \times \sqrt[10]{\frac{1}{16 x^{2}}} \\ \sqrt{2 x} = \sqrt[10]{{(2 x)}^{5}} = \sqrt[10]{2^{5} x^{5}} \\ \sqrt[5]{4 x} = \sqrt[10]{{(2^{2} x)}^{2}} = \sqrt[10]{2^{4} x^{2}} \\ \sqrt{2 x} \times \sqrt[5]{4 x} \times \sqrt[10]{\frac{1}{16 x^{2}}} & = \sqrt[10]{2^{5} x^{5}} \times \sqrt[10]{2^{4} x^{2}} \times \sqrt[10]{\frac{1}{2^{4} x^{2}}} \\ = \sqrt[10]{\frac{2^{5} x^{5} \times 2^{4} x^{2}}{2^{4} x^{2}}} \\ = \sqrt[10]{2^{5} x^{5}} \\ = {(2 x)}^{5} \\ = \sqrt{2 x} \end{matrix}$
$\begin{matrix} \frac{2}{3} \sqrt{\frac{2 b}{a}} \times \frac{3}{8} \sqrt[3]{\frac{a^{2}}{4 b^{2}}} \\ \frac{2}{3} \sqrt{\frac{2 b}{a}} = \frac{2}{3} \sqrt{\frac{2 b}{a} \times \frac{a}{a}} = \frac{2}{3 a} \sqrt{2 a b} = \frac{2}{3 a} \sqrt[6]{{(2 a b)}^{3}} = \frac{2}{3 a} \sqrt[6]{2^{3} a^{3} b^{3}} \\ \frac{3}{8} \sqrt[3]{\frac{a^{2}}{4 b^{2}}} = \frac{3}{8} \sqrt[3]{\frac{a^{2}}{2^{2} b^{2}} \times \frac{2 b}{2 b}} = \frac{3}{16 b} \sqrt[3]{2 a^{2} b} = \frac{3}{16 b} \sqrt[6]{{(2 a^{2} b)}^{2}} = \frac{3}{2^{4} b} \sqrt[6]{2^{2} a^{4} b^{2}} \\ \frac{2}{3} \sqrt{\frac{2 b}{a}} \times \frac{3}{8} \sqrt[3]{\frac{a^{2}}{4 b^{2}}} & = \frac{2}{3 a} \sqrt[6]{2^{3} a^{3} b^{3}} \times \frac{3}{2^{} b} \sqrt[6]{2^{2} a^{4} b^{2}} \\ = \frac{1}{2^{3} a b} \sqrt[6]{2^{5} a^{7} b^{5}} \\ = \frac{1}{2^{3} a b} a \sqrt[6]{2^{5} a b^{5}} \\ = \frac{1}{8 b} \sqrt[6]{32 a b^{5}} \end{matrix}$
$\begin{matrix} \frac{1}{2} \sqrt{\frac{1}{3}} \times \frac{3}{2} \sqrt[3]{\frac{1}{9}} \times \sqrt[6]{243} \\ \frac{1}{2} \sqrt{\frac{1}{3}} = \frac{1}{2} \sqrt[6]{{(\frac{1}{3})}^{3}} = \frac{1}{2} \sqrt[6]{\frac{1}{3^{3}}} \\ \frac{3}{2} \sqrt[3]{\frac{1}{9}} = \frac{3}{2} \sqrt[6]{{(\frac{1}{3^{2}})}^{2}} = \frac{3}{2} \sqrt[6]{\frac{1}{3^{4}}} \\ \frac{1}{2} \sqrt{\frac{1}{3}} \times \frac{3}{2} \sqrt[3]{\frac{1}{9}} \times \sqrt[6]{243} & = \frac{1}{2} \sqrt[6]{\frac{1}{3^{3}}} \times \frac{3}{2} \sqrt[6]{\frac{1}{3^{4}}} \times \sqrt[6]{3^{5}} \\ = \frac{3}{4} \sqrt[6]{\frac{3^{5}}{3^{} \times 3^{4}}} \\ = \frac{3}{4} \frac{1}{3^{2}} \\ = \frac{3}{4} \sqrt[3]{\frac{1}{3}} \end{matrix}$