L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{55}{9}-\dfrac{39}{8}-\left(\left(\left(0\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{4}\right)\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{18}\right)\) et \( Y=\left(\left(\dfrac{63}{8}\right)\sqrt{18}+\left(0\right)\sqrt{4}+\left(-\dfrac{38}{7}\right)\sqrt{50}+\dfrac{9}{8}\right)-\left(\left(\dfrac{26}{7}\right)\sqrt{8}+\left(-\dfrac{58}{7}\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{4}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{55}{9}-\dfrac{39}{8}-\left(\left(\left(0\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{4}\right)\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{18}\right)\right)+\left(\left(\left(\dfrac{63}{8}\right)\sqrt{18}+\left(0\right)\sqrt{4}+\left(-\dfrac{38}{7}\right)\sqrt{50}+\dfrac{9}{8}\right)-\left(\left(\dfrac{26}{7}\right)\sqrt{8}+\left(-\dfrac{58}{7}\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{4}\right)\right)\\
&=&\left(\dfrac{55}{9}-\dfrac{39}{8}-\left(\left(\left(0\right)\sqrt{2}\right)-19\right)-\left(\left(-53\right)\sqrt{2}\right)\right)+\left(\left(\left(\dfrac{189}{8}\right)\sqrt{2}+0+\left(-\dfrac{190}{7}\right)\sqrt{2}+\dfrac{9}{8}\right)-\left(\left(\dfrac{52}{7}\right)\sqrt{2}+\left(-\dfrac{290}{7}\right)\sqrt{2}+25\right)\right)\\
&=&\dfrac{55}{9}-\dfrac{39}{8}-\left(\left(\left(0\right)\sqrt{2}\right)-19\right)-\left(\left(-53\right)\sqrt{2}\right)+\left(\left(\dfrac{189}{8}\right)\sqrt{2}+0+\left(-\dfrac{190}{7}\right)\sqrt{2}+\dfrac{9}{8}\right)-\left(\left(\dfrac{52}{7}\right)\sqrt{2}+\left(-\dfrac{290}{7}\right)\sqrt{2}+25\right)\\
&=&-\dfrac{131}{36}+\left(\dfrac{4675}{56}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{55}{9}-\dfrac{39}{8}-\left(\left(\left(0\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{4}\right)\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{18}\right)\right)-\left(\left(\left(\dfrac{63}{8}\right)\sqrt{18}+\left(0\right)\sqrt{4}+\left(-\dfrac{38}{7}\right)\sqrt{50}+\dfrac{9}{8}\right)-\left(\left(\dfrac{26}{7}\right)\sqrt{8}+\left(-\dfrac{58}{7}\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{4}\right)\right)\\
&=&\left(\dfrac{55}{9}-\dfrac{39}{8}-\left(\left(\left(0\right)\sqrt{2}\right)-19\right)-\left(\left(-53\right)\sqrt{2}\right)\right)-\left(\left(\left(\dfrac{189}{8}\right)\sqrt{2}+0+\left(-\dfrac{190}{7}\right)\sqrt{2}+\dfrac{9}{8}\right)-\left(\left(\dfrac{52}{7}\right)\sqrt{2}+\left(-\dfrac{290}{7}\right)\sqrt{2}+25\right)\right)\\
&=&\left(\dfrac{1457}{72}+\left(53\right)\sqrt{2}\right)-\left(\left(\dfrac{1707}{56}\right)\sqrt{2}-\dfrac{191}{8}\right)\\
&=&\dfrac{1457}{72}+\left(53\right)\sqrt{2}+\left(-\dfrac{1707}{56}\right)\sqrt{2}+\dfrac{191}{8}\\
&=&\dfrac{397}{9}+\left(\dfrac{1261}{56}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{55}{9}-\dfrac{39}{8}-\left(\left(\left(0\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{4}\right)\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{18}\right)\right)\times\left(\left(\left(\dfrac{63}{8}\right)\sqrt{18}+\left(0\right)\sqrt{4}+\left(-\dfrac{38}{7}\right)\sqrt{50}+\dfrac{9}{8}\right)-\left(\left(\dfrac{26}{7}\right)\sqrt{8}+\left(-\dfrac{58}{7}\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{4}\right)\right)\\
&=&\left(\dfrac{55}{9}-\dfrac{39}{8}-\left(\left(\left(0\right)\sqrt{2}\right)-19\right)-\left(\left(-53\right)\sqrt{2}\right)\right)\times\left(\left(\left(\dfrac{189}{8}\right)\sqrt{2}+0+\left(-\dfrac{190}{7}\right)\sqrt{2}+\dfrac{9}{8}\right)-\left(\left(\dfrac{52}{7}\right)\sqrt{2}+\left(-\dfrac{290}{7}\right)\sqrt{2}+25\right)\right)\\
&=&\left(\dfrac{1457}{72}+\left(53\right)\sqrt{2}\right)\left(\left(\dfrac{1707}{56}\right)\sqrt{2}-\dfrac{191}{8}\right)\\
&=&\left(-\dfrac{871631}{1344}\right)\sqrt{2}-\dfrac{278287}{576}+\left(\dfrac{90471}{56}\right)\sqrt{4}\\
&=&\left(-\dfrac{871631}{1344}\right)\sqrt{2}+\dfrac{11079815}{4032}\\
\end{eqnarray*}