▶️ Ejercicio 285 álgebra - Solucionario Baldor ✅✅

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CAPITULO XXXVI

Transformación de radicales dobles

Ejercicio 285

Transformar en suma algebraica de radicales simples:

$\begin{matrix} \sqrt{5 + \sqrt{24}} \\ S e a : \\ a = 5 \\ b = 24 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{5^{2} - 24} \\ m & = \sqrt{25 - 24} \\ m & = \sqrt{1} \\ m & = 1 \\ \sqrt{5 + \sqrt{24}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{5 + 1}{2}} + \sqrt{\frac{5 - 1}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = \sqrt{3} + \sqrt{2} \end{matrix}$
$\begin{matrix} \sqrt{8 - \sqrt{60}} \\ S e a : \\ a = 8 \\ b = 60 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{8^{2} - 60} \\ m & = \sqrt{64 - 60} \\ m & = \sqrt{4} \\ m & = 2 \\ \sqrt{8 - \sqrt{60}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{8 + 2}{2}} - \sqrt{\frac{8 - 2}{2}} \\ = \sqrt{\frac{}{2}} - \sqrt{\frac{}{2}} \\ = \sqrt{5} - \sqrt{3} \end{matrix}$
$\begin{matrix} \sqrt{8 + \sqrt{28}} \\ S e a : \\ a = 8 \\ b = 28 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{8^{2} - 28} \\ m & = \sqrt{64 - 28} \\ m & = \sqrt{36} \\ m & = 6 \\ \sqrt{8 + \sqrt{28}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{8 + 6}{2}} + \sqrt{\frac{8 - 6}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{2}{2}} \\ = \sqrt{7} + 1 \end{matrix}$
$\begin{matrix} \sqrt{32 - \sqrt{700}} \\ S e a : \\ a = 32 \\ b = 700 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{3 2^{2} - 700} \\ m & = \sqrt{1024 - 700} \\ m & = \sqrt{324} \\ m & = 18 \\ \sqrt{32 - \sqrt{700}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{32 + 18}{2}} - \sqrt{\frac{32 - 18}{2}} \\ = \sqrt{\frac{}{2}} - \sqrt{\frac{}{2}} \\ = 5 - \sqrt{7} \end{matrix}$
$\begin{matrix} \sqrt{14 + \sqrt{132}} \\ S e a : \\ a = 14 \\ b = 132 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{1 4^{2} - 132} \\ m & = \sqrt{196 - 132} \\ m & = \sqrt{64} \\ m & = 8 \\ \sqrt{14 + \sqrt{132}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{14 + 8}{2}} + \sqrt{\frac{14 - 8}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = \sqrt{11} + \sqrt{3} \end{matrix}$
$\begin{matrix} \sqrt{13 + \sqrt{88}} \\ S e a : \\ a = 13 \\ b = 88 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{1 3^{2} - 88} \\ m & = \sqrt{169 - 88} \\ m & = \sqrt{81} \\ m & = 9 \\ \sqrt{13 + \sqrt{88}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{13 + 9}{2}} + \sqrt{\frac{13 - 9}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = \sqrt{11} + \sqrt{2} \end{matrix}$
$\begin{matrix} \sqrt{11 + 2 \sqrt{30}} & = \sqrt{11 + \sqrt{120}} \\ S e a : \\ a = 11 \\ b = 120 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{1 1^{2} - 120} \\ m & = \sqrt{121 - 120} \\ m & = \sqrt{1} \\ m & = 1 \\ \sqrt{11 + 2 \sqrt{30}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{11 + 1}{2}} + \sqrt{\frac{11 - 1}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = \sqrt{6} + \sqrt{5} \end{matrix}$
$\begin{matrix} \sqrt{84 - 18 \sqrt{3}} & = \sqrt{84 - \sqrt{972}} \\ S e a : \\ a = 84 \\ b = 972 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{8 4^{2} - 972} \\ m & = \sqrt{7056 - 972} \\ m & = \sqrt{6084} \\ m & = 78 \\ \sqrt{84 - 18 \sqrt{3}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{84 + 78}{2}} - \sqrt{\frac{84 - 78}{2}} \\ = \sqrt{\frac{}{2}} - \sqrt{\frac{}{2}} \\ = 9 - \sqrt{3} \end{matrix}$
$\begin{matrix} \sqrt{21 + 6 \sqrt{10}} & = \sqrt{21 + \sqrt{360}} \\ S e a : \\ a = 21 \\ b = 360 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{2 1^{2} - 360} \\ m & = \sqrt{441 - 360} \\ m & = \sqrt{81} \\ m & = 9 \\ \sqrt{21 + 6 \sqrt{10}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{21 + 9}{2}} + \sqrt{\frac{21 - 9}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = \sqrt{15} + \sqrt{6} \end{matrix}$
$\begin{matrix} \sqrt{28 + 14 \sqrt{3}} & = \sqrt{28 + \sqrt{588}} \\ S e a : \\ a = 28 \\ b = 588 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{2 8^{2} - 588} \\ m & = \sqrt{784 - 588} \\ m & = \sqrt{196} \\ m & = 14 \\ \sqrt{28 + 14 \sqrt{3}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{28 + 14}{2}} + \sqrt{\frac{28 - 14}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = \sqrt{21} + \sqrt{7} \end{matrix}$
$\begin{matrix} \sqrt{14 - 4 \sqrt{6}} & = \sqrt{14 - \sqrt{96}} \\ S e a : \\ a = 14 \\ b = 96 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{1 4^{2} - 96} \\ m & = \sqrt{196 - 96} \\ m & = \sqrt{100} \\ m & = 10 \\ \sqrt{14 - 4 \sqrt{6}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{14 + 10}{2}} - \sqrt{\frac{14 - 10}{2}} \\ = \sqrt{\frac{}{2}} - \sqrt{\frac{}{2}} \\ = \sqrt{12} - \sqrt{2} \\ = 2 \sqrt{3} - \sqrt{2} \end{matrix}$
$\begin{matrix} \sqrt{55 + 30 \sqrt{2}} & = \sqrt{55 + \sqrt{1800}} \\ S e a : \\ a = 55 \\ b = 1800 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{5 5^{2} - 1800} \\ m & = \sqrt{3025 - 1800} \\ m & = \sqrt{1225} \\ m & = 35 \\ \sqrt{55 + 30 \sqrt{2}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{55 + 35}{2}} + \sqrt{\frac{55 - 35}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = \sqrt{45} + \sqrt{10} \\ = \sqrt{5. 3^{2}} + \sqrt{10} \\ = 3 \sqrt{5} + \sqrt{10} \end{matrix}$
$\begin{matrix} \sqrt{73 - 12 \sqrt{35}} & = \sqrt{73 - \sqrt{5040}} \\ S e a : \\ a = 73 \\ b = 5040 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{7 3^{2} - 5040} \\ m & = \sqrt{5329 - 5040} \\ m & = \sqrt{289} \\ m & = 17 \\ \sqrt{73 - 12 \sqrt{35}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{73 + 17}{2}} - \sqrt{\frac{73 - 17}{2}} \\ = \sqrt{\frac{}{2}} - \sqrt{\frac{}{2}} \\ = \sqrt{45} - \sqrt{28} \\ = \sqrt{5. 3^{2}} - \sqrt{7. 2^{2}} \\ = 3 \sqrt{5} - 2 \sqrt{7} \end{matrix}$
$\begin{matrix} \sqrt{253 - 60 \sqrt{7}} & = \sqrt{253 - \sqrt{25200}} \\ S e a : \\ a = 253 \\ b = 25200 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{25 3^{2} - 25200} \\ m & = \sqrt{64009 - 25200} \\ m & = \sqrt{38809} \\ m & = 197 \\ \sqrt{253 - 60 \sqrt{7}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{253 + 197}{2}} - \sqrt{\frac{253 - 197}{2}} \\ = \sqrt{\frac{}{2}} - \sqrt{\frac{}{2}} \\ = 15 - \sqrt{28} \\ = 15 - \sqrt{7. 2^{2}} \\ = 15 - 2 \sqrt{7} \end{matrix}$
$\begin{matrix} \sqrt{293 - 30 \sqrt{22}} & = \sqrt{293 - \sqrt{19800}} \\ S e a : \\ a = 293 \\ b = 19800 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{29 3^{2} - 19800} \\ m & = \sqrt{85849 - 19800} \\ m & = \sqrt{66049} \\ m & = 257 \\ \sqrt{293 - 30 \sqrt{22}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{293 + 257}{2}} - \sqrt{\frac{293 - 257}{2}} \\ = \sqrt{\frac{}{2}} - \sqrt{\frac{}{2}} \\ = \sqrt{275} - \sqrt{18} \\ = \sqrt{11. 5^{2}} - \sqrt{2. 3^{2}} \\ = 5 \sqrt{11} - 3 \sqrt{2} \end{matrix}$
$\begin{matrix} \sqrt{\frac{5}{6} + \sqrt{\frac{2}{3}}} \\ S e a : \\ a = \frac{5}{6} \\ b = \frac{2}{3} \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{{(\frac{5}{6})}^{2} - \frac{2}{3}} \\ m & = \sqrt{\frac{25}{36} - \frac{2}{3}} \\ m & = \sqrt{\frac{1}{36}} \\ m & = \frac{1}{6} \\ \sqrt{\frac{5}{6} + \sqrt{\frac{2}{3}}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{\frac{5}{6} + \frac{1}{6}}{2}} + \sqrt{\frac{\frac{5}{6} - \frac{1}{6}}{2}} \\ = \sqrt{\frac{1}{2}} + \sqrt{\frac{\frac{4}{}}{2}} \\ = \sqrt{\frac{1}{2}} + \sqrt{\frac{1}{3}} \\ = \frac{\sqrt{1}}{\sqrt{2}} \frac{\sqrt{2}}{\sqrt{2}} + \frac{\sqrt{1}}{\sqrt{3}} \frac{\sqrt{3}}{\sqrt{3}} \\ = \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{3} \end{matrix}$
$\begin{matrix} \sqrt{\frac{3}{4} - \sqrt{\frac{1}{2}}} \\ S e a : \\ a = \frac{3}{4} \\ b = \frac{1}{2} \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{{(\frac{3}{4})}^{2} - \frac{1}{2}} \\ m & = \sqrt{\frac{9}{16} - \frac{1}{2}} \\ m & = \sqrt{\frac{1}{16}} \\ m & = \frac{1}{4} \\ \sqrt{\frac{3}{4} - \sqrt{\frac{1}{2}}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{\frac{3}{4} + \frac{1}{4}}{2}} - \sqrt{\frac{\frac{3}{4} - \frac{1}{4}}{2}} \\ = \sqrt{\frac{1}{2}} - \sqrt{\frac{\frac{1}{2}}{2}} \\ = \sqrt{\frac{1}{2}} - \sqrt{\frac{1}{4}} \\ = \frac{\sqrt{1}}{\sqrt{2}} \frac{\sqrt{2}}{\sqrt{2}} - \frac{1}{2} \\ = \frac{\sqrt{2}}{2} - \frac{1}{2} \\ = \frac{1}{2} (\sqrt{2} - 1) \end{matrix}$
$\begin{matrix} \sqrt{\frac{9}{16} + \sqrt{\frac{1}{8}}} \\ S e a : \\ a = \frac{9}{16} \\ b = \frac{1}{8} \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{{(\frac{9}{16})}^{2} - \frac{1}{8}} \\ m & = \sqrt{\frac{81}{256} - \frac{1}{8}} \\ m & = \sqrt{\frac{49}{256}} \\ m & = \frac{7}{16} \\ \sqrt{\frac{9}{16} + \sqrt{\frac{1}{8}}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{\frac{9}{16} + \frac{7}{16}}{2}} + \sqrt{\frac{\frac{9}{16} - \frac{7}{16}}{2}} \\ = \sqrt{\frac{1}{2}} + \sqrt{\frac{\frac{1}{8}}{2}} \\ = \sqrt{\frac{1}{2}} + \sqrt{\frac{1}{16}} \\ = \frac{\sqrt{1}}{\sqrt{2}} \frac{\sqrt{2}}{\sqrt{2}} + \frac{1}{4} \\ = \frac{\sqrt{2}}{2} + \frac{1}{4} \end{matrix}$
Hallar la raíz cuadrada de:
$\begin{matrix} 6 + 4 \sqrt{2} \\ \sqrt{6 + 4 \sqrt{2}} & = \sqrt{6 + \sqrt{32}} \\ S e a : \\ a = 6 \\ b = 32 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{6^{2} - 32} \\ m & = \sqrt{36 - 32} \\ m & = \sqrt{4} \\ m & = 2 \\ \sqrt{6 + 4 \sqrt{2}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{6 + 2}{2}} + \sqrt{\frac{6 - 2}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = 2 + \sqrt{2} \end{matrix}$
$\begin{matrix} 7 + 4 \sqrt{3} \\ \sqrt{7 + 4 \sqrt{3}} & = \sqrt{7 + \sqrt{48}} \\ S e a : \\ a = 7 \\ b = 48 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{7^{2} - 48} \\ m & = \sqrt{49 - 48} \\ m & = \sqrt{1} \\ m & = 1 \\ \sqrt{7 + 4 \sqrt{3}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{7 + 1}{2}} + \sqrt{\frac{7 - 1}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = 2 + \sqrt{3} \end{matrix}$
$\begin{matrix} 8 + 2 \sqrt{7} \\ \sqrt{8 + 2 \sqrt{7}} & = \sqrt{8 + \sqrt{28}} \\ S e a : \\ a = 8 \\ b = 28 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{8^{2} - 28} \\ m & = \sqrt{64 - 28} \\ m & = \sqrt{36} \\ m & = 6 \\ \sqrt{8 + 2 \sqrt{7}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{8 + 6}{2}} + \sqrt{\frac{8 - 6}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{2}{2}} \\ = \sqrt{7} + 1 \end{matrix}$
$\begin{matrix} 10 + 2 \sqrt{21} \\ \sqrt{10 + 2 \sqrt{21}} & = \sqrt{10 + \sqrt{84}} \\ S e a : \\ a = 10 \\ b = 84 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{1 0^{2} - 84} \\ m & = \sqrt{100 - 84} \\ m & = \sqrt{16} \\ m & = 4 \\ \sqrt{10 + 2 \sqrt{21}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{10 + 4}{2}} + \sqrt{\frac{10 - 4}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = \sqrt{7} + \sqrt{3} \end{matrix}$
$\begin{matrix} 18 + 6 \sqrt{5} \\ \sqrt{18 + 6 \sqrt{5}} & = \sqrt{18 + \sqrt{180}} \\ S e a : \\ a = 18 \\ b = 180 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{1 8^{2} - 180} \\ m & = \sqrt{324 - 180} \\ m & = \sqrt{144} \\ m & = 12 \\ \sqrt{18 + 6 \sqrt{5}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{18 + 12}{2}} + \sqrt{\frac{18 - 12}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = \sqrt{15} + \sqrt{3} \end{matrix}$
$\begin{matrix} 24 - 2 \sqrt{143} \\ \sqrt{24 - 2 \sqrt{143}} & = \sqrt{24 - \sqrt{572}} \\ S e a : \\ a = 24 \\ b = 572 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{2 4^{2} - 572} \\ m & = \sqrt{576 - 572} \\ m & = \sqrt{4} \\ m & = 2 \\ \sqrt{24 - 2 \sqrt{143}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{24 + 2}{2}} - \sqrt{\frac{24 - 2}{2}} \\ = \sqrt{\frac{}{2}} - \sqrt{\frac{}{2}} \\ = \sqrt{13} - \sqrt{11} \end{matrix}$
$\begin{matrix} 30 - 20 \sqrt{2} \\ \sqrt{30 - 20 \sqrt{2}} & = \sqrt{30 - \sqrt{800}} \\ S e a : \\ a = 30 \\ b = 800 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{3 0^{2} - 800} \\ m & = \sqrt{900 - 800} \\ m & = \sqrt{100} \\ m & = 10 \\ \sqrt{30 - 20 \sqrt{2}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{30 + 10}{2}} - \sqrt{\frac{30 - 10}{2}} \\ = \sqrt{\frac{}{2}} - \sqrt{\frac{}{2}} \\ = \sqrt{20} - \sqrt{10} \\ = \sqrt{5. 2^{2}} - \sqrt{10} \\ = 2 \sqrt{5} - \sqrt{10} \end{matrix}$
$\begin{matrix} 9 + 6 \sqrt{2} \\ \sqrt{9 + 6 \sqrt{2}} & = \sqrt{9 + \sqrt{72}} \\ S e a : \\ a = 9 \\ b = 72 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{9^{2} - 72} \\ m & = \sqrt{81 - 72} \\ m & = \sqrt{9} \\ m & = 3 \\ \sqrt{9 + 6 \sqrt{2}} & = \sqrt{\frac{a + m}{2}} + \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{9 + 3}{2}} + \sqrt{\frac{9 - 3}{2}} \\ = \sqrt{\frac{}{2}} + \sqrt{\frac{}{2}} \\ = \sqrt{6} + \sqrt{3} \end{matrix}$
$\begin{matrix} 98 - 24 \sqrt{5} \\ \sqrt{98 - 24 \sqrt{5}} & = \sqrt{98 - \sqrt{2880}} \\ S e a : \\ a = 98 \\ b = 2880 \\ m & = \sqrt{a^{2} - b} \\ m & = \sqrt{9 8^{2} - 2880} \\ m & = \sqrt{9604 - 2880} \\ m & = \sqrt{6724} \\ m & = 82 \\ \sqrt{98 - 24 \sqrt{5}} & = \sqrt{\frac{a + m}{2}} - \sqrt{\frac{a - m}{2}} \\ = \sqrt{\frac{98 + 82}{2}} - \sqrt{\frac{98 - 82}{2}} \\ = \sqrt{\frac{}{2}} - \sqrt{\frac{}{2}} \\ = \sqrt{90} - \sqrt{8} \\ = \sqrt{10. 3^{2}} - \sqrt{2. 2^{2}} \\ = 3 \sqrt{10} - 2 \sqrt{2} \end{matrix}$

Categories: Capítulo XXXVI