Ejercicio 249 – Solucionario Baldor

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CAPITULO XXXI

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

Ejercicio 249

Racionalizar el denominador de:

$\begin{matrix} \frac{\sqrt{3}}{\sqrt{2} + \sqrt{3} - \sqrt{5}} & = \frac{\sqrt{3}}{(\sqrt{2} + \sqrt{3}) - \sqrt{5}} \times \frac{\sqrt{2} + \sqrt{3} + \sqrt{5}}{(\sqrt{2} + \sqrt{3}) + \sqrt{5}} \\ = \frac{\sqrt{3} (\sqrt{2} + \sqrt{3} + \sqrt{5})}{{(\sqrt{2} + \sqrt{3})}^{2} - {(\sqrt{5})}^{2}} \\ = \frac{\sqrt{6} + \sqrt{3^{2}} + \sqrt{15}}{2 + 2 \sqrt{6} + 3 - 5} \\ = \frac{\sqrt{6} + \sqrt{15} + 3}{2 \sqrt{6}} \\ = \frac{\sqrt{6} + \sqrt{15} + 3}{2 \sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} \\ = \frac{\sqrt{6^{2}} + \sqrt{10. 3^{2}} + 3 \sqrt{6}}{2 (6)} \\ = \frac{6 + 3 \sqrt{10} + 3 \sqrt{6}}{2 (6)} \\ = \frac{3 (2 + \sqrt{10} + \sqrt{6})}{2} \\ = \frac{2 + \sqrt{10} + \sqrt{6}}{4} \end{matrix}$
$\begin{matrix} \frac{\sqrt{2}}{\sqrt{2} + \sqrt{3} + \sqrt{6}} & = \frac{\sqrt{2}}{(\sqrt{2} + \sqrt{3}) + \sqrt{6}} \times \frac{\sqrt{2} + \sqrt{3} - \sqrt{6}}{(\sqrt{2} + \sqrt{3}) - \sqrt{6}} \\ = \frac{\sqrt{2} (\sqrt{2} + \sqrt{3} - \sqrt{6})}{{(\sqrt{2} + \sqrt{3})}^{2} - {(\sqrt{6})}^{2}} \\ = \frac{\sqrt{2^{2}} + \sqrt{6} - \sqrt{3. 2^{2}}}{2 + 2 \sqrt{6} + 3 - 6} \\ = \frac{2 + \sqrt{6} - 2 \sqrt{3}}{2 \sqrt{6} - 1} \times \frac{2 \sqrt{6} + 1}{2 \sqrt{6} + 1} \\ = \frac{4 \sqrt{6} + 2 (6) - 4 \sqrt{18} + 2 + \sqrt{6} - 2 \sqrt{3}}{{(2 \sqrt{6})}^{2} - 1} \\ = \frac{5 \sqrt{6} + 14 - 4 \sqrt{2. 3^{3}} - 2 \sqrt{3}}{4 (6) - 1} \\ = \frac{5 \sqrt{6} - 12 \sqrt{2} - 2 \sqrt{3} + 14}{23} \end{matrix}$
$\begin{matrix} \frac{2 - \sqrt{3}}{2 + \sqrt{3} + \sqrt{5}} & = \frac{2 - \sqrt{3}}{(2 + \sqrt{3}) + \sqrt{5}} \times \frac{2 + \sqrt{3} - \sqrt{5}}{(2 + \sqrt{3}) - \sqrt{5}} \\ = \frac{4 + 2 \sqrt{3} - 2 \sqrt{5} - 2 \sqrt{3} - \sqrt{3^{2}} + \sqrt{15}}{{(2 + \sqrt{3})}^{2} - {(\sqrt{5})}^{2}} \\ = \frac{4 - 2 \sqrt{5} - 3 + \sqrt{15}}{2^{2} + 4 \sqrt{3} + 3 - 5} \\ = \frac{\sqrt{15} - 2 \sqrt{5} + 1}{4 \sqrt{3} + 2} \times \frac{4 \sqrt{3} - 2}{4 \sqrt{3} - 2} \\ = \frac{4 \sqrt{45} - 8 \sqrt{15} + 4 \sqrt{3} - 2 \sqrt{15} + 4 \sqrt{5} - 2}{{(4 \sqrt{3})}^{2} - 2^{2}} \\ = \frac{4 \sqrt{5. 3^{3}} - 10 \sqrt{15} + 4 \sqrt{3} + 4 \sqrt{5} - 2}{16 (3) - 4} \\ = \frac{12 \sqrt{5} - 10 \sqrt{15} + 4 \sqrt{3} + 4 \sqrt{5} - 2}{48 - 4} \\ = \frac{16 \sqrt{5} - 10 \sqrt{15} + 4 \sqrt{3} - 2}{44} \\ = \frac{2 (8 \sqrt{5} - 5 \sqrt{15} + 2 \sqrt{3} - 1)}{} \\ = \frac{8 \sqrt{5} - 5 \sqrt{15} + 2 \sqrt{3} - 1}{22} \end{matrix}$
$\begin{matrix} \frac{\sqrt{3} + \sqrt{5}}{\sqrt{2} + \sqrt{3} + \sqrt{5}} & = \frac{\sqrt{3} + \sqrt{5}}{(\sqrt{2} + \sqrt{3}) + \sqrt{5}} \times \frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{(\sqrt{2} + \sqrt{3}) - \sqrt{5}} \\ = \frac{\sqrt{6} + \sqrt{3^{2}} - \sqrt{15} + \sqrt{10} + \sqrt{15} - \sqrt{5^{2}}}{{(\sqrt{2} + \sqrt{3})}^{2} - {(\sqrt{5})}^{2}} \\ = \frac{\sqrt{6} + 3 + \sqrt{10} - 5}{2 + 2 \sqrt{6} + 3 - 5} \\ = \frac{\sqrt{6} + \sqrt{10} - 2}{2 \sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} \\ = \frac{\sqrt{6^{2}} + \sqrt{60} - 2 \sqrt{6}}{2 (6)} \\ = \frac{6 + \sqrt{2^{2} .15} - 2 \sqrt{6}}{2 (6)} \\ = \frac{6 + 2 \sqrt{15} - 2 \sqrt{6}}{2 (6)} \\ = \frac{2 (3 + \sqrt{15} - \sqrt{6})}{2 (6)} \\ = \frac{3 + \sqrt{15} - \sqrt{6}}{6} \end{matrix}$
$\begin{matrix} \frac{\sqrt{6} + \sqrt{3} + \sqrt{2}}{\sqrt{6} + \sqrt{3} - \sqrt{2}} & = \frac{\sqrt{6} + \sqrt{3} + \sqrt{2}}{(\sqrt{6} + \sqrt{3}) - \sqrt{2}} \times \frac{\sqrt{6} + \sqrt{3} + \sqrt{2}}{(\sqrt{6} + \sqrt{3}) + \sqrt{2}} \\ = \frac{{(\sqrt{6} + \sqrt{3} + \sqrt{2})}^{2}}{{(\sqrt{6} + \sqrt{3})}^{2} - {(\sqrt{2})}^{2}} \\ = \frac{6 + 3 + 2 + 2 \sqrt{18} + 2 \sqrt{12} + 2 \sqrt{6}}{6 + 2 \sqrt{18} + 3 - 2} \\ = \frac{11 + 2 \sqrt{2. 3^{2}} + 2 \sqrt{3. 2^{2}} + 2 \sqrt{6}}{7 + 2 \sqrt{2. 3^{2}}} \\ = \frac{11 + 6 \sqrt{2} + 4 \sqrt{3} + 2 \sqrt{6}}{7 + 6 \sqrt{2}} \times \frac{7 - 6 \sqrt{2}}{7 - 6 \sqrt{2}} \\ = \frac{77 + 42 \sqrt{2} + 28 \sqrt{3} + 14 \sqrt{6} - 66 \sqrt{2} - 36 \sqrt{2^{2}} - 24 \sqrt{6} - 12 \sqrt{12}}{7^{2} - {(6 \sqrt{2})}^{2}} \\ = \frac{77 + 28 \sqrt{3} - 24 \sqrt{2} - 36 (2) - 10 \sqrt{6} - 12 \sqrt{3. 2^{2}}}{49 - 36 (2)} \\ = \frac{77 + 28 \sqrt{3} - 24 \sqrt{2} - 72 - 10 \sqrt{6} - 24 \sqrt{3}}{49 - 72} \\ = \frac{5 + 4 \sqrt{3} - 24 \sqrt{2} - 10 \sqrt{6}}{- 23} \\ = \frac{10 \sqrt{6} - 4 \sqrt{3} + 24 \sqrt{2} - 5}{23} \end{matrix}$
$\begin{matrix} \frac{\sqrt{2} - \sqrt{5}}{\sqrt{2} + \sqrt{5} - \sqrt{10}} & = \frac{\sqrt{2} - \sqrt{5}}{(\sqrt{2} + \sqrt{5}) - \sqrt{10}} \times \frac{\sqrt{2} + \sqrt{5} + \sqrt{10}}{(\sqrt{2} + \sqrt{5}) + \sqrt{10}} \\ = \frac{2 + \sqrt{10} + \sqrt{20} - \sqrt{10} - 5 - \sqrt{50}}{{(\sqrt{2} + \sqrt{5})}^{2} - {(\sqrt{10})}^{2}} \\ = \frac{- 3 + \sqrt{5. 2^{2}} - \sqrt{5^{2} .2}}{2 + 2 \sqrt{10} + 5 - 10} \\ = \frac{- 3 + 2 \sqrt{5} - 5 \sqrt{2}}{2 \sqrt{10} - 3} \times \frac{2 \sqrt{10} + 3}{2 \sqrt{10} + 3} \\ = \frac{- 6 \sqrt{10} + 4 \sqrt{50} - 10 \sqrt{20} - 9 + 6 \sqrt{5} - 15 \sqrt{2}}{{(2 \sqrt{10})}^{2} - 3^{2}} \\ = \frac{- 6 \sqrt{10} + 4 \sqrt{5^{2} .2} - 10 \sqrt{5. 2^{2}} - 9 + 6 \sqrt{5} - 15 \sqrt{2}}{4 (10) - 9} \\ = \frac{- 6 \sqrt{10} + 20 \sqrt{2} - 20 \sqrt{5} - 9 + 6 \sqrt{5} - 15 \sqrt{2}}{31} \\ = \frac{5 \sqrt{2} - 6 \sqrt{10} - 14 \sqrt{5} - 9}{31} \end{matrix}$