▶️ Ejercicio 235 álgebra - Solucionario Baldor ✅✅

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CAPITULO XXXI

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Reducción al mínimo común índice

Ejercicio 235

Reducir al mínimo común índice:

$\begin{matrix} \sqrt{5}, \sqrt[3]{2} \\ m . c . m = 6 \\ \sqrt{5} = 5^{\frac{1}{2}} = 5^{\frac{3}{6}} = \sqrt[6]{5^{3}} = \sqrt[6]{75} \\ \sqrt[3]{2} = 2^{\frac{1}{3}} = 2^{\frac{2}{6}} = \sqrt[6]{2^{2}} = \sqrt[6]{4} \end{matrix}$
$\begin{matrix} \sqrt{2}, \sqrt[4]{3} \\ m . c . m = 4 \\ \sqrt{2} = 2^{\frac{1}{2}} = 2^{\frac{2}{4}} = \sqrt[4]{2^{2}} = \sqrt[4]{4} \\ \sqrt[4]{3} \end{matrix}$
$\begin{matrix} \sqrt{3,} \sqrt[3]{4}, \sqrt[4]{8} \\ m . c . m = 12 \\ \sqrt{3} = 3^{\frac{1}{2}} = 3^{\frac{6}{12}} = \sqrt[12]{3^{6}} = \sqrt[12]{729} \\ \sqrt[3]{4} = 4^{\frac{1}{3}} = 4^{\frac{4}{12}} = \sqrt[12]{4^{4}} = \sqrt[12]{256} \\ \sqrt[4]{8} = 8^{\frac{1}{4}} = 8^{\frac{3}{12}} = \sqrt[12]{8^{3}} = \sqrt[12]{512} \end{matrix}$
$\begin{matrix} \sqrt{2}, \sqrt[3]{3}, \sqrt[4]{5}, \sqrt[6]{7} \\ m . c . m = 12 \\ \sqrt{2} = 2^{\frac{1}{2}} = 2^{\frac{6}{12}} = \sqrt[12]{2^{6}} = \sqrt[12]{64} \\ \sqrt[3]{3} = 3^{\frac{1}{3}} = 3^{\frac{4}{12}} = \sqrt[12]{3^{4}} = \sqrt[12]{81} \\ \sqrt[4]{5} = 5^{\frac{1}{4}} = 5^{\frac{3}{12}} = \sqrt[12]{5^{3}} = \sqrt[12]{75} \\ \sqrt[6]{7} = 7^{\frac{1}{6}} = 7^{\frac{2}{12}} = \sqrt[12]{7^{2}} = \sqrt[12]{49} \end{matrix}$
$\begin{matrix} \sqrt{5 x}, \sqrt[3]{4 x^{2} y}, \sqrt[6]{7 a^{3} b} \\ m . c . m = 6 \\ \sqrt{5 x} = {(5 x)}^{\frac{1}{2}} = {(5 x)}^{\frac{3}{6}} = \sqrt[6]{{(5 x)}^{3}} = \sqrt[6]{75 x^{3}} \\ \sqrt[3]{4 x^{2} y} = {(4 x^{2} y)}^{\frac{1}{3}} = {(4 x^{2} y)}^{\frac{2}{6}} = \sqrt[6]{{(4 x^{2} y)}^{2}} = \sqrt[6]{16 x^{4} y^{2}} \\ \sqrt[6]{7 a^{3} b} \end{matrix}$
$\begin{matrix} \sqrt[3]{2 a b}, \sqrt[5]{3 a^{2} x}, \sqrt[15]{5 a^{3} x^{2}} \\ m . c . m = 15 \\ \sqrt[3]{2 a b} = {(2 a b)}^{\frac{1}{3}} = {(2 a b)}^{\frac{5}{15}} = \sqrt[15]{{(2 a b)}^{5}} = \sqrt[15]{2^{5} a^{5} b^{5}} = \sqrt[15]{32 a^{5} b^{5}} \\ \sqrt[5]{3 a^{2} x} = {(3 a^{2} x)}^{\frac{1}{5}} = {(3 a^{2} x)}^{\frac{3}{15}} = \sqrt[15]{{(3 a^{2} x)}^{3}} = \sqrt[15]{3^{3} a^{6} x^{3}} = \sqrt[15]{27 a^{6} x^{3}} \\ \sqrt[15]{5 a^{3} x^{2}} \end{matrix}$
$\begin{matrix} \sqrt[4]{8 a^{2} x^{3}}, \sqrt[6]{3 a^{5} m^{4}} \\ m . c . m = 12 \\ \sqrt[4]{8 a^{2} x^{3}} = {(8 a^{2} x^{3})}^{\frac{1}{4}} = {(8 a^{2} x^{3})}^{\frac{3}{12}} = \sqrt[12]{{(8 a^{2} x^{3})}^{3}} = \sqrt[12]{8^{3} a^{6} x^{9}} = \sqrt[12]{512 a^{6} x^{9}} \\ \sqrt[6]{3 a^{5} m^{4}} = {(3 a^{5} m^{4})}^{\frac{1}{6}} = {(3 a^{5} m^{4})}^{\frac{2}{12}} = \sqrt[12]{{(3 a^{5} m^{4})}^{2}} = \sqrt[12]{3^{2} a^{10} m^{8}} = \sqrt[12]{9 a^{10} m^{8}} \end{matrix}$
$\begin{matrix} \sqrt[3]{x^{2}}, \sqrt[6]{2 y^{3}}, \sqrt[9]{5 m^{7}} \\ m . c . m = 18 \\ \sqrt[3]{x^{2}} = {(x^{2})}^{\frac{1}{3}} = {(x^{2})}^{\frac{6}{18}} = \sqrt[18]{{(x^{2})}^{6}} = \sqrt[18]{x^{12}} \\ \sqrt[6]{2 y^{3}} = {(2 y^{3})}^{\frac{1}{6}} = {(2 y^{3})}^{\frac{3}{18}} = \sqrt[18]{{(2 y^{3})}^{3}} = \sqrt[18]{2^{3} y^{9}} = \sqrt[18]{8 y^{9}} \\ \sqrt[9]{5 m^{7}} = {(5 m^{7})}^{\frac{1}{9}} = {(5 m^{7})}^{\frac{2}{18}} = \sqrt[18]{{(5 m^{7})}^{2}} = \sqrt[18]{5^{2} m^{14}} = \sqrt[18]{25 m^{14}} \end{matrix}$
$\begin{matrix} \sqrt[4]{3 a}, \sqrt[5]{2 b^{2}}, \sqrt[10]{7 x^{3}} \\ m . c . m = 20 \\ \sqrt[4]{3 a} = {(3 a)}^{\frac{1}{4}} = {(3 a)}^{\frac{5}{20}} = \sqrt[20]{{(3 a)}^{5}} = \sqrt[20]{3^{5} a^{5}} = \sqrt[20]{243 a^{5}} \\ \sqrt[5]{2 b^{2}} = {(2 b^{2})}^{\frac{1}{5}} = {(2 b^{2})}^{\frac{4}{20}} = \sqrt[20]{{(2 b^{2})}^{4}} = \sqrt[20]{2^{4} b^{8}} = \sqrt[20]{16 b^{8}} \\ \sqrt[10]{7 x^{3}} = {(7 x^{3})}^{\frac{1}{10}} = {(7 x^{3})}^{\frac{2}{10}} = \sqrt[20]{{(7 x^{3})}^{2}} = \sqrt[20]{7^{2} x^{6}} = \sqrt[20]{49 x^{6}} \end{matrix}$
$\begin{matrix} 2 \sqrt[3]{a}, 3 \sqrt{2 b}, 4 \sqrt[4]{5 x^{2}} \\ m . c . m = 12 \\ 2 \sqrt[3]{a} = 2 a^{\frac{1}{3}} = 2 a^{\frac{4}{12}} = 2 \sqrt[12]{a^{4}} \\ 3 \sqrt{2 b} = 3 {(2 b)}^{\frac{1}{2}} = 3 {(2 b)}^{\frac{6}{12}} = 3 \sqrt[12]{{(2 b)}^{6}} = 3 \sqrt[12]{2^{6} b^{6}} = 3 \sqrt[12]{64 b^{6}} \\ 4 \sqrt[4]{5 x^{2}} = 4 {(5 x^{2})}^{\frac{1}{4}} = 4 {(5 x^{2})}^{\frac{3}{12}} = 4 \sqrt[12]{{(5 x^{2})}^{3}} = 4 \sqrt[12]{5^{3} x^{6}} = 4 \sqrt[12]{75 x^{6}} \end{matrix}$
$\begin{matrix} 3 \sqrt[3]{a^{2}}, \frac{1}{2} \sqrt[6]{b^{3}}, 4 \sqrt[9]{x^{5}} \\ m . c . m = 18 \\ 3 \sqrt[3]{a^{2}} = 3 {(a^{2})}^{\frac{1}{3}} = 3 {(a^{2})}^{\frac{6}{18}} = 3 \sqrt[18]{a^{12}} \\ \frac{1}{2} \sqrt[6]{b^{3}} = \frac{1}{2} {(b^{3})}^{\frac{1}{6}} = \frac{1}{2} {(b^{3})}^{\frac{3}{18}} = \frac{1}{2} \sqrt[18]{b^{9}} \\ 4 \sqrt[9]{x^{5}} = 4 {(x^{5})}^{\frac{1}{9}} = 4 {(x^{5})}^{\frac{2}{18}} = 4 \sqrt[18]{x^{10}} \end{matrix}$
$\begin{matrix} \sqrt{2 m}, 3 \sqrt[5]{a^{3} x^{4}}, 2 \sqrt[10]{x^{7} y^{2}} \\ m . c . m = 10 \\ \sqrt{2 m} = {(2 m)}^{\frac{1}{2}} = {(2 m)}^{\frac{5}{10}} = \sqrt[10]{{(2 m)}^{5}} = \sqrt[10]{2^{5} m^{5}} = \sqrt[10]{32 m^{5}} \\ 3 \sqrt[5]{a^{3} x^{4}} = 3 {(a^{3} x^{4})}^{\frac{1}{5}} = 3 {(a^{3} x^{4})}^{\frac{2}{10}} = 3 \sqrt[10]{{(a^{3} x^{4})}^{2}} = 3 \sqrt[10]{a^{6} x^{8}} \\ 2 \sqrt[10]{x^{7} y^{2}} \end{matrix}$

Categories: Capítulo XXXI