Ejercicio 248 – Solucionario Baldor

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CAPITULO XXXI

R a d i c a l e s
Racionalización de expresiones conjugadas

Ejercicio 248

Racionalizar el denominador de:

$\begin{matrix} \frac{3 - \sqrt{2}}{1 + \sqrt{2}} & = \frac{3 - \sqrt{2}}{1 + \sqrt{2}} \times \frac{1 - \sqrt{2}}{1 - \sqrt{2}} \\ = \frac{3 - \sqrt{2} - 3 \sqrt{2} + \sqrt{2^{2}}}{1 - \sqrt{2^{2}}} \\ = \frac{3 - 4 \sqrt{2} + 2}{1 - 2} \\ = \frac{5 - 4 \sqrt{2}}{- 1} \\ = 4 \sqrt{2} - 5 \end{matrix}$
$\begin{matrix} \frac{5 + 2 \sqrt{3}}{4 - \sqrt{3}} & = \frac{5 + 2 \sqrt{3}}{4 - \sqrt{3}} \times \frac{4 + \sqrt{3}}{4 + \sqrt{3}} \\ = \frac{20 + 8 \sqrt{3} + 5 \sqrt{3} + 2 \sqrt{3^{2}}}{4^{2} - \sqrt{3^{2}}} \\ = \frac{20 + 13 \sqrt{3} + 2 (3)}{16 - 3} \\ = \frac{13 \sqrt{3} + 26}{13} \\ = \frac{13 (\sqrt{3} + 2)}{13} \\ = \sqrt{3} + 2 \end{matrix}$
$\begin{matrix} \frac{\sqrt{2} - \sqrt{5}}{\sqrt{2} + \sqrt{5}} & = \frac{\sqrt{2} - \sqrt{5}}{\sqrt{2} + \sqrt{5}} \times \frac{\sqrt{2} - \sqrt{5}}{\sqrt{2} - \sqrt{5}} \\ = \frac{{(\sqrt{2} - \sqrt{5})}^{2}}{\sqrt{2^{2}} - \sqrt{5^{2}}} \\ = \frac{\sqrt{2^{2}} - 2 \sqrt{2 \times 5} + \sqrt{5^{2}}}{2 - 5} \\ = \frac{2 - 2 \sqrt{10} + 5}{- 3} \\ = \frac{7 - 2 \sqrt{10}}{- 3} \\ = \frac{2 \sqrt{10} - 7}{3} \end{matrix}$
$\begin{matrix} \frac{\sqrt{7} + 2 \sqrt{5}}{\sqrt{7} - \sqrt{5}} & = \frac{\sqrt{7} + 2 \sqrt{5}}{\sqrt{7} - \sqrt{5}} \times \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}} \\ = \frac{\sqrt{7^{2}} + 2 \sqrt{35} + \sqrt{35} + 2 \sqrt{5^{2}}}{\sqrt{7^{2}} - \sqrt{5^{2}}} \\ = \frac{7 + 3 \sqrt{35} + 2 (5)}{7 - 5} \\ = \frac{17 + 3 \sqrt{35}}{2} \end{matrix}$
$\begin{matrix} \frac{\sqrt{2} - 3 \sqrt{5}}{2 \sqrt{2} + \sqrt{5}} & = \frac{\sqrt{2} - 3 \sqrt{5}}{2 \sqrt{2} + \sqrt{5}} \times \frac{2 \sqrt{2} - \sqrt{5}}{2 \sqrt{2} - \sqrt{5}} \\ = \frac{2 \sqrt{2^{2}} - 6 \sqrt{10} - \sqrt{10} + 3 \sqrt{5^{2}}}{{(2 \sqrt{2})}^{2} - \sqrt{5^{2}}} \\ = \frac{2 (2) - 7 \sqrt{10} + 3 (5)}{4 (2) - 5} \\ = \frac{4 - 7 \sqrt{10} + 15}{8 - 5} \\ = \frac{19 - 7 \sqrt{10}}{3} \end{matrix}$
$\begin{matrix} \frac{19}{5 \sqrt{2} - 4 \sqrt{3}} & = \frac{19}{5 \sqrt{2} - 4 \sqrt{3}} \times \frac{5 \sqrt{2} + 4 \sqrt{3}}{5 \sqrt{2} + 4 \sqrt{3}} \\ = \frac{19 (5 \sqrt{2} + 4 \sqrt{3})}{{(5 \sqrt{2})}^{2} - {(4 \sqrt{3})}^{2}} \\ = \frac{19 (5 \sqrt{2} + 4 \sqrt{3})}{25 (2) - 16 (3)} \\ = \frac{19 (5 \sqrt{2} + 4 \sqrt{3})}{50 - 48} \\ = \frac{19 (5 \sqrt{2} + 4 \sqrt{3})}{2} \end{matrix}$
$\begin{matrix} \frac{3 \sqrt{2}}{7 \sqrt{2} - 6 \sqrt{3}} & = \frac{3 \sqrt{2}}{7 \sqrt{2} - 6 \sqrt{3}} \times \frac{7 \sqrt{2} + 6 \sqrt{3}}{7 \sqrt{2} + 6 \sqrt{3}} \\ = \frac{21 \sqrt{2^{2}} + 18 \sqrt{6}}{{(7 \sqrt{2})}^{2} - {(6 \sqrt{3})}^{2}} \\ = \frac{21 (2) + 18 \sqrt{6}}{49 (2) - 36 (3)} \\ = \frac{42 + 18 \sqrt{6}}{98 - 108} \\ = \frac{2 (21 + 9 \sqrt{6})}{-} \\ = - \frac{21 + 9 \sqrt{6}}{5} \end{matrix}$
$\begin{matrix} \frac{4 \sqrt{3} - 3 \sqrt{7}}{2 \sqrt{3} + 3 \sqrt{7}} & = \frac{4 \sqrt{3} - 3 \sqrt{7}}{2 \sqrt{3} + 3 \sqrt{7}} \times \frac{2 \sqrt{3} - 3 \sqrt{7}}{2 \sqrt{3} - 3 \sqrt{7}} \\ = \frac{8 \sqrt{3^{2}} - 6 \sqrt{21} - 12 \sqrt{21} + 9 \sqrt{7^{2}}}{{(2 \sqrt{3})}^{2} - {(3 \sqrt{7})}^{2}} \\ = \frac{8 (3) - 18 \sqrt{21} + 9 (7)}{4 (3) - 9 (7)} \\ = \frac{24 - 18 \sqrt{21} + 63}{12 - 63} \\ = \frac{87 - 18 \sqrt{21}}{- 51} \\ = \frac{18 \sqrt{21} - 87}{51} \\ = \frac{3 (6 \sqrt{21} - 29)}{} \\ = \frac{6 \sqrt{21} - 29}{17} \end{matrix}$
$\begin{matrix} \frac{5 \sqrt{2} - 6 \sqrt{3}}{4 \sqrt{2} - 3 \sqrt{3}} & = \frac{5 \sqrt{2} - 6 \sqrt{3}}{4 \sqrt{2} - 3 \sqrt{3}} \times \frac{4 \sqrt{2} + 3 \sqrt{3}}{4 \sqrt{2} + 3 \sqrt{3}} \\ = \frac{20 \sqrt{2^{2}} - 24 \sqrt{6} + 15 \sqrt{6} - 18 \sqrt{3^{2}}}{{(4 \sqrt{2})}^{2} - {(3 \sqrt{3})}^{2}} \\ = \frac{20 (2) - 9 \sqrt{6} - 18 (3)}{16 (2) - 9 (3)} \\ = \frac{40 - 9 \sqrt{6} - 54}{32 - 27} \\ = \frac{40 - 9 \sqrt{6} - 54}{32 - 27} \\ = - \frac{14 + 9 \sqrt{6}}{5} \end{matrix}$
$\begin{matrix} \frac{\sqrt{7} + 3 \sqrt{11}}{5 \sqrt{7} + 4 \sqrt{11}} & = \frac{\sqrt{7} + 3 \sqrt{11}}{5 \sqrt{7} + 4 \sqrt{11}} \times \frac{5 \sqrt{7} - 4 \sqrt{11}}{5 \sqrt{7} - 4 \sqrt{11}} \\ = \frac{5 \sqrt{7^{2}} + 15 \sqrt{77} - 4 \sqrt{77} - 12 \sqrt{1 1^{2}}}{{(5 \sqrt{7})}^{2} - {(4 \sqrt{11})}^{2}} \\ = \frac{5 (7) + 11 \sqrt{77} - 12 (11)}{25 (7) - 16 (11)} \\ = \frac{35 + 11 \sqrt{77} - 132}{175 - 176} \\ = 97 - 11 \sqrt{77} \end{matrix}$
$\begin{matrix} \frac{\sqrt{5} + \sqrt{2}}{7 + 2 \sqrt{10}} & = \frac{\sqrt{5} + \sqrt{2}}{7 + 2 \sqrt{10}} \times \frac{7 - 2 \sqrt{10}}{7 - 2 \sqrt{10}} \\ = \frac{7 \sqrt{5} + 7 \sqrt{2} - 2 \sqrt{50} - 2 \sqrt{20}}{7^{2} - {(2 \sqrt{10})}^{2}} \\ = \frac{7 \sqrt{5} + 7 \sqrt{2} - 2 \sqrt{5^{2} .2} - 2 \sqrt{5. 2^{2}}}{49 - 4 (10)} \\ = \frac{7 \sqrt{5} + 7 \sqrt{2} - 10 \sqrt{2} - 4 \sqrt{5}}{9} \\ = \frac{3 \sqrt{5} - 3 \sqrt{2}}{9} \\ = \frac{3 (\sqrt{5} - \sqrt{2})}{} \\ = \frac{\sqrt{5} - \sqrt{2}}{3} \end{matrix}$
$\begin{matrix} \frac{9 \sqrt{3} - 3 \sqrt{2}}{6 - \sqrt{6}} & = \frac{9 \sqrt{3} - 3 \sqrt{2}}{6 - \sqrt{6}} \times \frac{6 + \sqrt{6}}{6 + \sqrt{6}} \\ = \frac{54 \sqrt{3} - 18 \sqrt{2} + 9 \sqrt{18} - 3 \sqrt{12}}{6^{2} - \sqrt{6^{2}}} \\ = \frac{54 \sqrt{3} - 18 \sqrt{2} + 9 \sqrt{3^{2} .2} - 3 \sqrt{3. 2^{2}}}{36 - 6} \\ = \frac{54 \sqrt{3} - 18 \sqrt{2} + 27 \sqrt{2} - 6 \sqrt{3}}{30} \\ = \frac{48 \sqrt{3} + 9 \sqrt{2}}{30} \\ = \frac{3 (16 \sqrt{3} + 3 \sqrt{2})}{} \\ = \frac{16 \sqrt{3} + 3 \sqrt{2}}{10} \end{matrix}$
$\begin{matrix} \frac{\sqrt{a} + \sqrt{x}}{2 \sqrt{a} + \sqrt{x}} & = \frac{\sqrt{a} + \sqrt{x}}{2 \sqrt{a} + \sqrt{x}} \times \frac{2 \sqrt{a} - \sqrt{x}}{2 \sqrt{a} - \sqrt{x}} \\ = \frac{2 \sqrt{a^{2}} + 2 \sqrt{a x} - \sqrt{a x} - \sqrt{x^{2}}}{{(2 \sqrt{a})}^{2} - {(\sqrt{x})}^{2}} \\ = \frac{2 a + \sqrt{a x} - x}{4 a - x} \end{matrix}$
$\begin{matrix} \frac{\sqrt{x} - \sqrt{x - 1}}{\sqrt{x} + \sqrt{x - 1}} & = \frac{\sqrt{x} - \sqrt{x - 1}}{\sqrt{x} + \sqrt{x - 1}} \times \frac{\sqrt{x} - \sqrt{x - 1}}{\sqrt{x} - \sqrt{x - 1}} \\ = \frac{\sqrt{x^{2}} - \sqrt{x (x - 1)} - \sqrt{x (x - 1)} + \sqrt{{(x - 1)}^{2}}}{\sqrt{x^{2}} - \sqrt{{(x - 1)}^{2}}} \\ = \frac{x - 2 \sqrt{x (x - 1)} + x - 1}{x - (x - 1)} \\ = \frac{2 x - 2 \sqrt{x (x - 1)} - 1}{x - x + 1} \\ = 2 x - 2 \sqrt{x (x - 1)} - 1 \end{matrix}$
$\begin{matrix} \frac{\sqrt{a} - \sqrt{a + 1}}{\sqrt{a} + \sqrt{a + 1}} & = \frac{\sqrt{a} - \sqrt{a + 1}}{\sqrt{a} + \sqrt{a + 1}} \times \frac{\sqrt{a} - \sqrt{a + 1}}{\sqrt{a} - \sqrt{a + 1}} \\ = \frac{{(\sqrt{a} - \sqrt{a + 1})}^{2}}{{(\sqrt{a})}^{2} - {(\sqrt{a + 1})}^{2}} \\ = \frac{a - 2 \sqrt{a (a + 1)} + a + 1}{a - (a + 1)} \\ = \frac{2 a - 2 \sqrt{a (a + 1)} + 1}{a - a - 1} \\ = 2 \sqrt{a (a + 1)} - 2 a - 1 \end{matrix}$
$\begin{matrix} \frac{\sqrt{x + 2} + \sqrt{2}}{\sqrt{x + 2} - \sqrt{2}} & = \frac{\sqrt{x + 2} + \sqrt{2}}{\sqrt{x + 2} - \sqrt{2}} \times \frac{\sqrt{x + 2} + \sqrt{2}}{\sqrt{x + 2} + \sqrt{2}} \\ = \frac{{(\sqrt{x + 2} + \sqrt{2})}^{2}}{{(\sqrt{x + 2})}^{2} - {(\sqrt{2})}^{2}} \\ = \frac{x + 2 + 2 \sqrt{2 (x + 2)} + 2}{x + 2 - 2} \\ = \frac{x + 2 \sqrt{2 (x + 2)} + 4}{x} \end{matrix}$
$\begin{matrix} \frac{\sqrt{a + 4} - \sqrt{a}}{\sqrt{a + 4} + \sqrt{a}} & = \frac{\sqrt{a + 4} - \sqrt{a}}{\sqrt{a + 4} + \sqrt{a}} \times \frac{\sqrt{a + 4} - \sqrt{a}}{\sqrt{a + 4} - \sqrt{a}} \\ = \frac{{(\sqrt{a + 4} - \sqrt{a})}^{2}}{{(\sqrt{a + 4})}^{2} - {(\sqrt{a})}^{2}} \\ = \frac{a + 4 - 2 \sqrt{a (a + 4)} + a}{a + 4 - a} \\ = \frac{2 a - 2 \sqrt{a (a + 4)} + 4}{4} \\ = \frac{2 (a - \sqrt{a (a + 4)} + 2)}{} \\ = \frac{a - \sqrt{a (a + 4)} + 2}{2} \end{matrix}$
$\begin{matrix} \frac{\sqrt{a + b} - \sqrt{a - b}}{\sqrt{a + b} + \sqrt{a - b}} & = \frac{\sqrt{a + b} - \sqrt{a - b}}{\sqrt{a + b} + \sqrt{a - b}} \times \frac{\sqrt{a + b} - \sqrt{a - b}}{\sqrt{a + b} - \sqrt{a - b}} \\ = \frac{{(\sqrt{a + b} - \sqrt{a - b})}^{2}}{{(\sqrt{a + b})}^{2} - {(\sqrt{a - b})}^{2}} \\ = \frac{a + b - 2 \sqrt{a^{2} - b^{2}} + a - b}{a + b - (a - b)} \\ = \frac{2 a - 2 \sqrt{a^{2} - b^{2}}}{a + b - a + b} \\ = \frac{2 (a - \sqrt{a^{2} - b^{2}})}{2 b} \\ = \frac{a - \sqrt{a^{2} - b^{2}}}{b} \end{matrix}$