L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{2}{3}-\left(\left(\dfrac{37}{9}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{64}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{46}{7}\right)\sqrt{18}\right)\right)-\left(\left(-9\right)\sqrt{50}\right)\) et \( Y=\left(-\dfrac{65}{6}\right)\sqrt{18}+\left(\dfrac{13}{6}\right)\sqrt{4}+\left(\dfrac{71}{4}\right)\sqrt{8}+\left(-\dfrac{29}{2}\right)\sqrt{50}+\left(\dfrac{25}{3}\right)\sqrt{8}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{2}{3}-\left(\left(\dfrac{37}{9}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{64}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{46}{7}\right)\sqrt{18}\right)\right)-\left(\left(-9\right)\sqrt{50}\right)\right)+\left(\left(-\dfrac{65}{6}\right)\sqrt{18}+\left(\dfrac{13}{6}\right)\sqrt{4}+\left(\dfrac{71}{4}\right)\sqrt{8}+\left(-\dfrac{29}{2}\right)\sqrt{50}+\left(\dfrac{25}{3}\right)\sqrt{8}\right)\\
&=&\left(\dfrac{2}{3}-\left(\left(\dfrac{185}{9}\right)\sqrt{2}\right)-\left(\dfrac{128}{9}-\left(\left(-\dfrac{64}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{138}{7}\right)\sqrt{2}\right)\right)-\left(\left(-45\right)\sqrt{2}\right)\right)+\left(\left(-\dfrac{65}{2}\right)\sqrt{2}+\dfrac{13}{3}+\left(\dfrac{71}{2}\right)\sqrt{2}+\left(-\dfrac{145}{2}\right)\sqrt{2}+\left(\dfrac{50}{3}\right)\sqrt{2}\right)\\
&=&\dfrac{2}{3}-\left(\left(\dfrac{185}{9}\right)\sqrt{2}\right)-\left(\dfrac{128}{9}-\left(\left(-\dfrac{64}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{138}{7}\right)\sqrt{2}\right)\right)-\left(\left(-45\right)\sqrt{2}\right)+\left(-\dfrac{65}{2}\right)\sqrt{2}+\dfrac{13}{3}+\left(\dfrac{71}{2}\right)\sqrt{2}+\left(-\dfrac{145}{2}\right)\sqrt{2}+\left(\dfrac{50}{3}\right)\sqrt{2}\\
&=&-\dfrac{83}{9}+\left(-\dfrac{3781}{126}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{2}{3}-\left(\left(\dfrac{37}{9}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{64}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{46}{7}\right)\sqrt{18}\right)\right)-\left(\left(-9\right)\sqrt{50}\right)\right)-\left(\left(-\dfrac{65}{6}\right)\sqrt{18}+\left(\dfrac{13}{6}\right)\sqrt{4}+\left(\dfrac{71}{4}\right)\sqrt{8}+\left(-\dfrac{29}{2}\right)\sqrt{50}+\left(\dfrac{25}{3}\right)\sqrt{8}\right)\\
&=&\left(\dfrac{2}{3}-\left(\left(\dfrac{185}{9}\right)\sqrt{2}\right)-\left(\dfrac{128}{9}-\left(\left(-\dfrac{64}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{138}{7}\right)\sqrt{2}\right)\right)-\left(\left(-45\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{65}{2}\right)\sqrt{2}+\dfrac{13}{3}+\left(\dfrac{71}{2}\right)\sqrt{2}+\left(-\dfrac{145}{2}\right)\sqrt{2}+\left(\dfrac{50}{3}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{122}{9}+\left(\dfrac{1438}{63}\right)\sqrt{2}\right)-\left(\left(-\dfrac{317}{6}\right)\sqrt{2}+\dfrac{13}{3}\right)\\
&=&-\dfrac{122}{9}+\left(\dfrac{1438}{63}\right)\sqrt{2}+\left(\dfrac{317}{6}\right)\sqrt{2}-\dfrac{13}{3}\\
&=&-\dfrac{161}{9}+\left(\dfrac{9533}{126}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{2}{3}-\left(\left(\dfrac{37}{9}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{64}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{46}{7}\right)\sqrt{18}\right)\right)-\left(\left(-9\right)\sqrt{50}\right)\right)\times\left(\left(-\dfrac{65}{6}\right)\sqrt{18}+\left(\dfrac{13}{6}\right)\sqrt{4}+\left(\dfrac{71}{4}\right)\sqrt{8}+\left(-\dfrac{29}{2}\right)\sqrt{50}+\left(\dfrac{25}{3}\right)\sqrt{8}\right)\\
&=&\left(\dfrac{2}{3}-\left(\left(\dfrac{185}{9}\right)\sqrt{2}\right)-\left(\dfrac{128}{9}-\left(\left(-\dfrac{64}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{138}{7}\right)\sqrt{2}\right)\right)-\left(\left(-45\right)\sqrt{2}\right)\right)\times\left(\left(-\dfrac{65}{2}\right)\sqrt{2}+\dfrac{13}{3}+\left(\dfrac{71}{2}\right)\sqrt{2}+\left(-\dfrac{145}{2}\right)\sqrt{2}+\left(\dfrac{50}{3}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{122}{9}+\left(\dfrac{1438}{63}\right)\sqrt{2}\right)\left(\left(-\dfrac{317}{6}\right)\sqrt{2}+\dfrac{13}{3}\right)\\
&=&\left(\dfrac{17117}{21}\right)\sqrt{2}-\dfrac{1586}{27}+\left(-\dfrac{227923}{189}\right)\sqrt{4}\\
&=&\left(\dfrac{17117}{21}\right)\sqrt{2}-\dfrac{466948}{189}\\
\end{eqnarray*}