▶️ Ejercicio 231 álgebra - Solucionario Baldor ✅✅

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CAPITULO XXXI

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

Ejercicio 231

Simplificar:

$\sqrt{18} = \sqrt{3^{2} .2} = 3 \sqrt{2}$
$3 \sqrt{48} = 3 \sqrt{3. 4^{2}} = 3.4 \sqrt{3} = 12 \sqrt{3}$
$\sqrt[3]{16} = \sqrt[3]{2. 2^{3}} = 2 \sqrt[3]{2}$
$\frac{1}{2} \sqrt[3]{128} = \frac{1}{2} \sqrt[3]{4^{3} .2} = \frac{1}{2} \sqrt[3]{2} = 2 \sqrt[3]{2}$
$2 \sqrt[4]{243} = 2 \sqrt[4]{3^{4} .3} = 2.3 \sqrt[4]{3} = 6 \sqrt[4]{3}$
$\sqrt{50 a^{2} b} = \sqrt{2. 5^{2} a^{2} b} = 5 a \sqrt{2 b}$
$3 \sqrt{81 x^{3} y^{4}} = 3 \sqrt{9^{2} . x^{2} x . y^{4}} = 3.9 x y^{2} \sqrt{x} = 27 x y^{2} \sqrt{x}$
$\begin{matrix} \frac{1}{2} \sqrt{108 a^{5} b^{7}} & = \frac{1}{2} \sqrt{2^{2} . 3^{2} .3. a^{4} a . b^{6} b} \\ = \frac{2}{2} .3 a^{2} b^{3} \sqrt{3 a b} \\ = 3 a^{2} b^{3} \sqrt{3 a b} \end{matrix}$
$\frac{3}{5} \sqrt{125 m n^{6}} = \frac{3}{5} n^{3} \sqrt{5^{2} .5 m} = \frac{3. 5}{5} n^{3} \sqrt{5 m} = 3 n^{3} \sqrt{5 m}$
$\begin{matrix} 2 a \sqrt{44 a^{3} b^{7} c^{9}} & = 2 a \sqrt{11. 2^{2} a^{2} a b^{6} b c^{8} c} \\ = 2 a .2 a b^{3} c^{4} \sqrt{11 a b c} \\ = 4 a^{2} b^{3} c^{4} \sqrt{11 a b c} \end{matrix}$
$\begin{matrix} 2 \sqrt[3]{16 x^{2} y^{7}} & = 2 \sqrt[3]{2. 2^{3} x^{2} y^{6} y} \\ = 2.2 y^{2} \sqrt[3]{2 x^{2} y} \\ = 4 y^{2} \sqrt[3]{2 x^{2} y} \end{matrix}$
$\begin{matrix} \frac{2}{3} \sqrt[3]{27 m^{2} n^{8}} & = \frac{2}{3} \sqrt[3]{3^{3} . m^{2} n^{6} . n^{2}} \\ = \frac{2}{3} 3 n^{2} \sqrt[3]{m^{2} n^{2}} \\ = 2 n^{2} \sqrt[3]{m^{2} n^{2}} \end{matrix}$
$\begin{matrix} 5 a \sqrt[3]{160 x^{7} y^{9} z^{13}} & = 5 a \sqrt[3]{2^{3} .20 x^{6} x y^{9} z^{12} z} \\ = 5 a .2 x^{2} y^{3} z^{4} \sqrt[3]{20 x z} \\ = 10 a x^{2} y^{3} z^{4} \sqrt[3]{20 x z} \end{matrix}$
$\begin{matrix} \sqrt[4]{80 a^{4} b^{5} c^{12}} & = \sqrt[4]{2^{4} .5 a^{4} b^{4} b c^{12}} \\ = 2 a b c^{3} \sqrt[4]{5 b} \end{matrix}$
$\begin{matrix} 3 \sqrt[4]{5 x^{8} y^{14} z^{16}} & = 3 \sqrt[4]{5 x^{8} y^{12} y^{2} z^{16}} \\ = 3 x^{2} y^{3} z^{4} \sqrt[4]{5 y^{2}} \end{matrix}$
$\begin{matrix} \frac{2}{5} \sqrt[5]{32 x^{2} y^{11}} & = \frac{2}{5} \sqrt[5]{2^{5} x^{2} y^{10} y} \\ = \frac{2.2}{5} y^{2} \sqrt[5]{x^{2} y} \\ = \frac{4}{5} y^{2} \sqrt[5]{x^{2} y} \end{matrix}$
$\begin{matrix} 2 x y \sqrt[3]{128 x^{2} y^{8}} & = 2 x y \sqrt[3]{2^{6} .2 x^{2} y^{6} y^{2}} \\ = 2 x y (2^{2} y^{2}) \sqrt[3]{2 x^{2} y^{2}} \\ = 8 x y^{3} \sqrt[3]{2 x^{2} y^{2}} \end{matrix}$
$\begin{matrix} \frac{1}{3 a} \sqrt{27 a^{3} m^{7}} & = \frac{1}{3 a} \sqrt{3^{2} 3 a^{2} a m^{6} m} \\ = \frac{1}{3 a} 3 a m^{3} \sqrt{3 a m} \\ = m^{3} \sqrt{3 a m} \end{matrix}$
$\begin{matrix} \frac{3}{5 x} \sqrt[3]{375 a^{8} b} & = \frac{3}{5 x} \sqrt[3]{5^{3} .3 a^{6} a^{2} b} \\ = \frac{3}{5 x} 5 a^{2} \sqrt[3]{3 a^{2} b} \\ = \frac{3 a^{2}}{x} \sqrt[3]{3 a^{2} b} \end{matrix}$
$\begin{matrix} \frac{1}{3} \sqrt[4]{81 a^{4} b} & = \frac{1}{3} \sqrt[4]{3^{4} a^{4} b} \\ = \frac{1}{3} 3 a \sqrt[4]{b} \\ = a \sqrt[4]{b} \end{matrix}$
$\begin{matrix} \sqrt{9 a + 18 b} & = \sqrt{9 (a + 2 b)} \\ = \sqrt{3^{2} (a + 2 b)} \\ = 3 \sqrt{a + 2 b} \end{matrix}$
$\begin{matrix} \sqrt{3 a^{3} b^{2} - 3 a^{2} b^{2}} & = \sqrt{3 a^{2} b^{2} (a - 1)} \\ = a b \sqrt{3 (a - 1)} \end{matrix}$
$\begin{matrix} \sqrt{8 x^{2} y^{4} + 16 x y^{4}} & = \sqrt{8 x y^{4} (x + 2)} \\ = \sqrt{2^{2} .2 x y^{4} (x + 2)} \\ = 2 y^{2} \sqrt{2 x (x + 2)} \end{matrix}$
$\begin{matrix} \sqrt{2 x^{2} - 4 x y + 2 y^{2}} & = \sqrt{2 (x^{2} - 2 x y + y^{2})} \\ = \sqrt{2 {(x - y)}^{2}} \\ = \sqrt{2} (x - y) \end{matrix}$
$\begin{matrix} \sqrt{(a - b) (a^{2} - b^{2})} & = \sqrt{(a - b) (a - b) (a + b)} \\ = \sqrt{{(a - b)}^{2} (a + b)} \\ = (a - b) \sqrt{a + b} \end{matrix}$
$\begin{matrix} \sqrt{2 a m^{2} + 4 a m n + 2 a n^{2}} & = \sqrt{2 a (m^{2} + 2 m n + n^{2})} \\ = \sqrt{2 a {(m + n)}^{2}} \\ = \sqrt{2 a} (m + n) \end{matrix}$
$\begin{matrix} \sqrt{9 a^{3} - 36 a^{2} + 36 a} & = \sqrt{9 a (a^{2} - 4 a + 4)} \\ = \sqrt{3^{2} a {(a - 2)}^{2}} \\ = 3 (a - 2) \sqrt{a} \end{matrix}$

Categories: Capítulo XXXI