L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{58}{9}\right)\sqrt{25}+\left(-\dfrac{51}{4}\right)\sqrt{45}\) et \( Y=\left(-\dfrac{26}{7}\right)\sqrt{45}+\dfrac{3}{4}+\left(\dfrac{7}{3}\right)\sqrt{125}+\left(\left(4\right)\sqrt{45}\right)-\left(\left(-\dfrac{25}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{8}\right)\sqrt{25}\right)+\left(\left(\dfrac{35}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{25}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{58}{9}\right)\sqrt{25}+\left(-\dfrac{51}{4}\right)\sqrt{45}\right)+\left(\left(-\dfrac{26}{7}\right)\sqrt{45}+\dfrac{3}{4}+\left(\dfrac{7}{3}\right)\sqrt{125}+\left(\left(4\right)\sqrt{45}\right)-\left(\left(-\dfrac{25}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{8}\right)\sqrt{25}\right)+\left(\left(\dfrac{35}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{290}{9}+\left(-\dfrac{153}{4}\right)\sqrt{5}\right)+\left(\left(-\dfrac{78}{7}\right)\sqrt{5}+\dfrac{3}{4}+\left(\dfrac{35}{3}\right)\sqrt{5}+\left(\left(12\right)\sqrt{5}\right)-\left(\left(-\dfrac{125}{4}\right)\sqrt{5}\right)-\dfrac{185}{8}+\dfrac{175}{2}-\left(\left(-\dfrac{14}{3}\right)\sqrt{5}\right)-\dfrac{175}{2}\right)\\
&=&\dfrac{290}{9}+\left(-\dfrac{153}{4}\right)\sqrt{5}+\left(-\dfrac{78}{7}\right)\sqrt{5}+\dfrac{3}{4}+\left(\dfrac{35}{3}\right)\sqrt{5}+\left(\left(12\right)\sqrt{5}\right)-\left(\left(-\dfrac{125}{4}\right)\sqrt{5}\right)-\dfrac{185}{8}+\dfrac{175}{2}-\left(\left(-\dfrac{14}{3}\right)\sqrt{5}\right)-\dfrac{175}{2}\\
&=&\dfrac{709}{72}+\left(\dfrac{214}{21}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{58}{9}\right)\sqrt{25}+\left(-\dfrac{51}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{45}+\dfrac{3}{4}+\left(\dfrac{7}{3}\right)\sqrt{125}+\left(\left(4\right)\sqrt{45}\right)-\left(\left(-\dfrac{25}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{8}\right)\sqrt{25}\right)+\left(\left(\dfrac{35}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{290}{9}+\left(-\dfrac{153}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{78}{7}\right)\sqrt{5}+\dfrac{3}{4}+\left(\dfrac{35}{3}\right)\sqrt{5}+\left(\left(12\right)\sqrt{5}\right)-\left(\left(-\dfrac{125}{4}\right)\sqrt{5}\right)-\dfrac{185}{8}+\dfrac{175}{2}-\left(\left(-\dfrac{14}{3}\right)\sqrt{5}\right)-\dfrac{175}{2}\right)\\
&=&\left(\dfrac{290}{9}+\left(-\dfrac{153}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{4069}{84}\right)\sqrt{5}-\dfrac{179}{8}\right)\\
&=&\dfrac{290}{9}+\left(-\dfrac{153}{4}\right)\sqrt{5}+\left(-\dfrac{4069}{84}\right)\sqrt{5}+\dfrac{179}{8}\\
&=&\dfrac{3931}{72}+\left(-\dfrac{3641}{42}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{58}{9}\right)\sqrt{25}+\left(-\dfrac{51}{4}\right)\sqrt{45}\right)\times\left(\left(-\dfrac{26}{7}\right)\sqrt{45}+\dfrac{3}{4}+\left(\dfrac{7}{3}\right)\sqrt{125}+\left(\left(4\right)\sqrt{45}\right)-\left(\left(-\dfrac{25}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{8}\right)\sqrt{25}\right)+\left(\left(\dfrac{35}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{290}{9}+\left(-\dfrac{153}{4}\right)\sqrt{5}\right)\times\left(\left(-\dfrac{78}{7}\right)\sqrt{5}+\dfrac{3}{4}+\left(\dfrac{35}{3}\right)\sqrt{5}+\left(\left(12\right)\sqrt{5}\right)-\left(\left(-\dfrac{125}{4}\right)\sqrt{5}\right)-\dfrac{185}{8}+\dfrac{175}{2}-\left(\left(-\dfrac{14}{3}\right)\sqrt{5}\right)-\dfrac{175}{2}\right)\\
&=&\left(\dfrac{290}{9}+\left(-\dfrac{153}{4}\right)\sqrt{5}\right)\left(\left(\dfrac{4069}{84}\right)\sqrt{5}-\dfrac{179}{8}\right)\\
&=&\left(\dfrac{14616223}{6048}\right)\sqrt{5}-\dfrac{25955}{36}+\left(-\dfrac{207519}{112}\right)\sqrt{25}\\
&=&\left(\dfrac{14616223}{6048}\right)\sqrt{5}-\dfrac{10065095}{1008}\\
\end{eqnarray*}