L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{56}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{51}{8}\right)\sqrt{125}\right)-\left(\left(-\dfrac{71}{7}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{70}{9}\right)\sqrt{25}-\dfrac{7}{8}+\left(\dfrac{69}{2}\right)\sqrt{125}\right)\) et \( Y=\left(\dfrac{13}{4}\right)\sqrt{25}\) . 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(\dfrac{56}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{51}{8}\right)\sqrt{125}\right)-\left(\left(-\dfrac{71}{7}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{70}{9}\right)\sqrt{25}-\dfrac{7}{8}+\left(\dfrac{69}{2}\right)\sqrt{125}\right)\right)+\left(\left(\dfrac{13}{4}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\left(56\right)\sqrt{5}\right)-\left(\left(\dfrac{255}{8}\right)\sqrt{5}\right)-\left(\left(-\dfrac{213}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{350}{9}-\dfrac{7}{8}+\left(\dfrac{345}{2}\right)\sqrt{5}\right)\right)+\left(\dfrac{65}{4}\right)\\
&=&\left(\left(\left(56\right)\sqrt{5}\right)-\left(\left(\dfrac{255}{8}\right)\sqrt{5}\right)-\left(\left(-\dfrac{213}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{350}{9}-\dfrac{7}{8}+\left(\dfrac{345}{2}\right)\sqrt{5}\right)+\dfrac{65}{4}\\
&=&\left(-\dfrac{6605}{56}\right)\sqrt{5}-\dfrac{1567}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{56}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{51}{8}\right)\sqrt{125}\right)-\left(\left(-\dfrac{71}{7}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{70}{9}\right)\sqrt{25}-\dfrac{7}{8}+\left(\dfrac{69}{2}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{13}{4}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\left(56\right)\sqrt{5}\right)-\left(\left(\dfrac{255}{8}\right)\sqrt{5}\right)-\left(\left(-\dfrac{213}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{350}{9}-\dfrac{7}{8}+\left(\dfrac{345}{2}\right)\sqrt{5}\right)\right)-\left(\dfrac{65}{4}\right)\\
&=&\left(\left(-\dfrac{6605}{56}\right)\sqrt{5}-\dfrac{2737}{72}\right)-\left(\dfrac{65}{4}\right)\\
&=&\left(-\dfrac{6605}{56}\right)\sqrt{5}-\dfrac{2737}{72}+-\dfrac{65}{4}\\
&=&\left(-\dfrac{6605}{56}\right)\sqrt{5}-\dfrac{3907}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{56}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{51}{8}\right)\sqrt{125}\right)-\left(\left(-\dfrac{71}{7}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{70}{9}\right)\sqrt{25}-\dfrac{7}{8}+\left(\dfrac{69}{2}\right)\sqrt{125}\right)\right)\times\left(\left(\dfrac{13}{4}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\left(56\right)\sqrt{5}\right)-\left(\left(\dfrac{255}{8}\right)\sqrt{5}\right)-\left(\left(-\dfrac{213}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{350}{9}-\dfrac{7}{8}+\left(\dfrac{345}{2}\right)\sqrt{5}\right)\right)\times\left(\dfrac{65}{4}\right)\\
&=&\left(\left(-\dfrac{6605}{56}\right)\sqrt{5}-\dfrac{2737}{72}\right)\left(\dfrac{65}{4}\right)\\
&=&\left(-\dfrac{429325}{224}\right)\sqrt{5}-\dfrac{177905}{288}\\
&=&\left(-\dfrac{429325}{224}\right)\sqrt{5}-\dfrac{177905}{288}\\
\end{eqnarray*}