L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\left(5\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{75}+\left(-\dfrac{27}{5}\right)\sqrt{27}-\dfrac{65}{2}+\left(\dfrac{25}{7}\right)\sqrt{12}-\dfrac{77}{2}\right)-\left(-9+\dfrac{75}{7}+\left(\dfrac{13}{5}\right)\sqrt{27}-8\right)-\left(\left(\dfrac{17}{9}\right)\sqrt{12}+\left(\dfrac{1}{2}\right)\sqrt{9}+\left(\dfrac{66}{5}\right)\sqrt{9}+\left(\dfrac{25}{7}\right)\sqrt{75}+\left(\dfrac{13}{5}\right)\sqrt{27}\right)-\left(\dfrac{15}{7}+\left(\dfrac{64}{7}\right)\sqrt{9}+\left(6\right)\sqrt{12}\right)\) et \( Y=\dfrac{69}{8}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(5\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{75}+\left(-\dfrac{27}{5}\right)\sqrt{27}-\dfrac{65}{2}+\left(\dfrac{25}{7}\right)\sqrt{12}-\dfrac{77}{2}\right)-\left(-9+\dfrac{75}{7}+\left(\dfrac{13}{5}\right)\sqrt{27}-8\right)-\left(\left(\dfrac{17}{9}\right)\sqrt{12}+\left(\dfrac{1}{2}\right)\sqrt{9}+\left(\dfrac{66}{5}\right)\sqrt{9}+\left(\dfrac{25}{7}\right)\sqrt{75}+\left(\dfrac{13}{5}\right)\sqrt{27}\right)-\left(\dfrac{15}{7}+\left(\dfrac{64}{7}\right)\sqrt{9}+\left(6\right)\sqrt{12}\right)\right)+\left(\dfrac{69}{8}\right)\\
&=&\left(\left(\left(\left(15\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{3}+\left(-\dfrac{81}{5}\right)\sqrt{3}-\dfrac{65}{2}+\left(\dfrac{50}{7}\right)\sqrt{3}-\dfrac{77}{2}\right)-\left(-9+\dfrac{75}{7}+\left(\dfrac{39}{5}\right)\sqrt{3}-8\right)-\left(\left(\dfrac{34}{9}\right)\sqrt{3}+\dfrac{3}{2}+\dfrac{198}{5}+\left(\dfrac{125}{7}\right)\sqrt{3}+\left(\dfrac{39}{5}\right)\sqrt{3}\right)-\left(\dfrac{15}{7}+\dfrac{192}{7}+\left(12\right)\sqrt{3}\right)\right)+\left(\dfrac{69}{8}\right)\\
&=&\left(\left(\left(15\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{3}+\left(-\dfrac{81}{5}\right)\sqrt{3}-\dfrac{65}{2}+\left(\dfrac{50}{7}\right)\sqrt{3}-\dfrac{77}{2}\right)-\left(-9+\dfrac{75}{7}+\left(\dfrac{39}{5}\right)\sqrt{3}-8\right)-\left(\left(\dfrac{34}{9}\right)\sqrt{3}+\dfrac{3}{2}+\dfrac{198}{5}+\left(\dfrac{125}{7}\right)\sqrt{3}+\left(\dfrac{39}{5}\right)\sqrt{3}\right)-\left(\dfrac{15}{7}+\dfrac{192}{7}+\left(12\right)\sqrt{3}\right)+\dfrac{69}{8}\\
&=&\left(-\dfrac{2641}{90}\right)\sqrt{3}+\dfrac{4267}{280}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(5\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{75}+\left(-\dfrac{27}{5}\right)\sqrt{27}-\dfrac{65}{2}+\left(\dfrac{25}{7}\right)\sqrt{12}-\dfrac{77}{2}\right)-\left(-9+\dfrac{75}{7}+\left(\dfrac{13}{5}\right)\sqrt{27}-8\right)-\left(\left(\dfrac{17}{9}\right)\sqrt{12}+\left(\dfrac{1}{2}\right)\sqrt{9}+\left(\dfrac{66}{5}\right)\sqrt{9}+\left(\dfrac{25}{7}\right)\sqrt{75}+\left(\dfrac{13}{5}\right)\sqrt{27}\right)-\left(\dfrac{15}{7}+\left(\dfrac{64}{7}\right)\sqrt{9}+\left(6\right)\sqrt{12}\right)\right)-\left(\dfrac{69}{8}\right)\\
&=&\left(\left(\left(\left(15\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{3}+\left(-\dfrac{81}{5}\right)\sqrt{3}-\dfrac{65}{2}+\left(\dfrac{50}{7}\right)\sqrt{3}-\dfrac{77}{2}\right)-\left(-9+\dfrac{75}{7}+\left(\dfrac{39}{5}\right)\sqrt{3}-8\right)-\left(\left(\dfrac{34}{9}\right)\sqrt{3}+\dfrac{3}{2}+\dfrac{198}{5}+\left(\dfrac{125}{7}\right)\sqrt{3}+\left(\dfrac{39}{5}\right)\sqrt{3}\right)-\left(\dfrac{15}{7}+\dfrac{192}{7}+\left(12\right)\sqrt{3}\right)\right)-\left(\dfrac{69}{8}\right)\\
&=&\left(\left(-\dfrac{2641}{90}\right)\sqrt{3}+\dfrac{463}{70}\right)-\left(\dfrac{69}{8}\right)\\
&=&\left(-\dfrac{2641}{90}\right)\sqrt{3}+\dfrac{463}{70}+-\dfrac{69}{8}\\
&=&\left(-\dfrac{2641}{90}\right)\sqrt{3}-\dfrac{563}{280}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(5\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{75}+\left(-\dfrac{27}{5}\right)\sqrt{27}-\dfrac{65}{2}+\left(\dfrac{25}{7}\right)\sqrt{12}-\dfrac{77}{2}\right)-\left(-9+\dfrac{75}{7}+\left(\dfrac{13}{5}\right)\sqrt{27}-8\right)-\left(\left(\dfrac{17}{9}\right)\sqrt{12}+\left(\dfrac{1}{2}\right)\sqrt{9}+\left(\dfrac{66}{5}\right)\sqrt{9}+\left(\dfrac{25}{7}\right)\sqrt{75}+\left(\dfrac{13}{5}\right)\sqrt{27}\right)-\left(\dfrac{15}{7}+\left(\dfrac{64}{7}\right)\sqrt{9}+\left(6\right)\sqrt{12}\right)\right)\times\left(\dfrac{69}{8}\right)\\
&=&\left(\left(\left(\left(15\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{3}+\left(-\dfrac{81}{5}\right)\sqrt{3}-\dfrac{65}{2}+\left(\dfrac{50}{7}\right)\sqrt{3}-\dfrac{77}{2}\right)-\left(-9+\dfrac{75}{7}+\left(\dfrac{39}{5}\right)\sqrt{3}-8\right)-\left(\left(\dfrac{34}{9}\right)\sqrt{3}+\dfrac{3}{2}+\dfrac{198}{5}+\left(\dfrac{125}{7}\right)\sqrt{3}+\left(\dfrac{39}{5}\right)\sqrt{3}\right)-\left(\dfrac{15}{7}+\dfrac{192}{7}+\left(12\right)\sqrt{3}\right)\right)\times\left(\dfrac{69}{8}\right)\\
&=&\left(\left(-\dfrac{2641}{90}\right)\sqrt{3}+\dfrac{463}{70}\right)\left(\dfrac{69}{8}\right)\\
&=&\left(-\dfrac{60743}{240}\right)\sqrt{3}+\dfrac{31947}{560}\\
&=&\left(-\dfrac{60743}{240}\right)\sqrt{3}+\dfrac{31947}{560}\\
\end{eqnarray*}