L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{38}{3}\right)\sqrt{50}+\left(-\dfrac{63}{4}\right)\sqrt{18}+\left(\dfrac{69}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}+\left(-\dfrac{63}{4}\right)\sqrt{18}\right)-\dfrac{26}{7}\) et \( Y=\left(\left(-\dfrac{57}{2}\right)\sqrt{4}+\left(\dfrac{29}{7}\right)\sqrt{18}+\left(-\dfrac{4}{3}\right)\sqrt{8}+\left(-\dfrac{2}{7}\right)\sqrt{8}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{8}+\left(\dfrac{43}{5}\right)\sqrt{50}+\left(\dfrac{54}{7}\right)\sqrt{4}+\left(6\right)\sqrt{50}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\dfrac{38}{3}\right)\sqrt{50}+\left(-\dfrac{63}{4}\right)\sqrt{18}+\left(\dfrac{69}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}+\left(-\dfrac{63}{4}\right)\sqrt{18}\right)-\dfrac{26}{7}\right)+\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{4}+\left(\dfrac{29}{7}\right)\sqrt{18}+\left(-\dfrac{4}{3}\right)\sqrt{8}+\left(-\dfrac{2}{7}\right)\sqrt{8}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{8}+\left(\dfrac{43}{5}\right)\sqrt{50}+\left(\dfrac{54}{7}\right)\sqrt{4}+\left(6\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(\dfrac{190}{3}\right)\sqrt{2}+\left(-\dfrac{189}{4}\right)\sqrt{2}+\left(\dfrac{345}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}+\left(-\dfrac{189}{4}\right)\sqrt{2}\right)-\dfrac{26}{7}\right)+\left(\left(-57+\left(\dfrac{87}{7}\right)\sqrt{2}+\left(-\dfrac{8}{3}\right)\sqrt{2}+\left(-\dfrac{4}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{47}{4}\right)\sqrt{2}+\left(43\right)\sqrt{2}+\dfrac{108}{7}+\left(30\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{190}{3}\right)\sqrt{2}+\left(-\dfrac{189}{4}\right)\sqrt{2}+\left(\dfrac{345}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}+\left(-\dfrac{189}{4}\right)\sqrt{2}\right)-\dfrac{26}{7}+\left(-57+\left(\dfrac{87}{7}\right)\sqrt{2}+\left(-\dfrac{8}{3}\right)\sqrt{2}+\left(-\dfrac{4}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{47}{4}\right)\sqrt{2}+\left(43\right)\sqrt{2}+\dfrac{108}{7}+\left(30\right)\sqrt{2}\right)\\
&=&\left(\dfrac{89}{84}\right)\sqrt{2}-\dfrac{533}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{38}{3}\right)\sqrt{50}+\left(-\dfrac{63}{4}\right)\sqrt{18}+\left(\dfrac{69}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}+\left(-\dfrac{63}{4}\right)\sqrt{18}\right)-\dfrac{26}{7}\right)-\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{4}+\left(\dfrac{29}{7}\right)\sqrt{18}+\left(-\dfrac{4}{3}\right)\sqrt{8}+\left(-\dfrac{2}{7}\right)\sqrt{8}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{8}+\left(\dfrac{43}{5}\right)\sqrt{50}+\left(\dfrac{54}{7}\right)\sqrt{4}+\left(6\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(\dfrac{190}{3}\right)\sqrt{2}+\left(-\dfrac{189}{4}\right)\sqrt{2}+\left(\dfrac{345}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}+\left(-\dfrac{189}{4}\right)\sqrt{2}\right)-\dfrac{26}{7}\right)-\left(\left(-57+\left(\dfrac{87}{7}\right)\sqrt{2}+\left(-\dfrac{8}{3}\right)\sqrt{2}+\left(-\dfrac{4}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{47}{4}\right)\sqrt{2}+\left(43\right)\sqrt{2}+\dfrac{108}{7}+\left(30\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{2231}{42}\right)\sqrt{2}-\dfrac{26}{7}\right)-\left(-\dfrac{507}{7}+\left(-\dfrac{4373}{84}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2231}{42}\right)\sqrt{2}-\dfrac{26}{7}+\dfrac{507}{7}+\left(\dfrac{4373}{84}\right)\sqrt{2}\\
&=&\left(\dfrac{2945}{28}\right)\sqrt{2}+\dfrac{481}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{38}{3}\right)\sqrt{50}+\left(-\dfrac{63}{4}\right)\sqrt{18}+\left(\dfrac{69}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}+\left(-\dfrac{63}{4}\right)\sqrt{18}\right)-\dfrac{26}{7}\right)\times\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{4}+\left(\dfrac{29}{7}\right)\sqrt{18}+\left(-\dfrac{4}{3}\right)\sqrt{8}+\left(-\dfrac{2}{7}\right)\sqrt{8}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{8}+\left(\dfrac{43}{5}\right)\sqrt{50}+\left(\dfrac{54}{7}\right)\sqrt{4}+\left(6\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(\dfrac{190}{3}\right)\sqrt{2}+\left(-\dfrac{189}{4}\right)\sqrt{2}+\left(\dfrac{345}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}+\left(-\dfrac{189}{4}\right)\sqrt{2}\right)-\dfrac{26}{7}\right)\times\left(\left(-57+\left(\dfrac{87}{7}\right)\sqrt{2}+\left(-\dfrac{8}{3}\right)\sqrt{2}+\left(-\dfrac{4}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{47}{4}\right)\sqrt{2}+\left(43\right)\sqrt{2}+\dfrac{108}{7}+\left(30\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{2231}{42}\right)\sqrt{2}-\dfrac{26}{7}\right)\left(-\dfrac{507}{7}+\left(-\dfrac{4373}{84}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{537134}{147}\right)\sqrt{2}+\left(-\dfrac{9756163}{3528}\right)\sqrt{4}+\dfrac{13182}{49}\\
&=&\left(-\dfrac{537134}{147}\right)\sqrt{2}-\dfrac{9281611}{1764}\\
\end{eqnarray*}