L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{63}{2}-\left(\dfrac{23}{3}+\left(\dfrac{78}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{25}{9}\right)\sqrt{8}\right)\) et \( Y=\left(-3\right)\sqrt{50}+\left(-\dfrac{68}{5}\right)\sqrt{18}+\left(\dfrac{15}{4}\right)\sqrt{4}-\dfrac{54}{5}-\dfrac{74}{7}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{63}{2}-\left(\dfrac{23}{3}+\left(\dfrac{78}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{25}{9}\right)\sqrt{8}\right)\right)+\left(\left(-3\right)\sqrt{50}+\left(-\dfrac{68}{5}\right)\sqrt{18}+\left(\dfrac{15}{4}\right)\sqrt{4}-\dfrac{54}{5}-\dfrac{74}{7}\right)\\
&=&\left(\dfrac{63}{2}-\left(\dfrac{23}{3}+\left(\dfrac{390}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{50}{9}\right)\sqrt{2}\right)\right)+\left(\left(-15\right)\sqrt{2}+\left(-\dfrac{204}{5}\right)\sqrt{2}+\dfrac{15}{2}-\dfrac{54}{5}-\dfrac{74}{7}\right)\\
&=&\dfrac{63}{2}-\left(\dfrac{23}{3}+\left(\dfrac{390}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{50}{9}\right)\sqrt{2}\right)+\left(-15\right)\sqrt{2}+\left(-\dfrac{204}{5}\right)\sqrt{2}+\dfrac{15}{2}-\dfrac{54}{5}-\dfrac{74}{7}\\
&=&\dfrac{1046}{105}+\left(-\dfrac{36877}{315}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{63}{2}-\left(\dfrac{23}{3}+\left(\dfrac{78}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{25}{9}\right)\sqrt{8}\right)\right)-\left(\left(-3\right)\sqrt{50}+\left(-\dfrac{68}{5}\right)\sqrt{18}+\left(\dfrac{15}{4}\right)\sqrt{4}-\dfrac{54}{5}-\dfrac{74}{7}\right)\\
&=&\left(\dfrac{63}{2}-\left(\dfrac{23}{3}+\left(\dfrac{390}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{50}{9}\right)\sqrt{2}\right)\right)-\left(\left(-15\right)\sqrt{2}+\left(-\dfrac{204}{5}\right)\sqrt{2}+\dfrac{15}{2}-\dfrac{54}{5}-\dfrac{74}{7}\right)\\
&=&\left(\dfrac{143}{6}+\left(-\dfrac{3860}{63}\right)\sqrt{2}\right)-\left(\left(-\dfrac{279}{5}\right)\sqrt{2}-\dfrac{971}{70}\right)\\
&=&\dfrac{143}{6}+\left(-\dfrac{3860}{63}\right)\sqrt{2}+\left(\dfrac{279}{5}\right)\sqrt{2}+\dfrac{971}{70}\\
&=&\dfrac{3959}{105}+\left(-\dfrac{1723}{315}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{63}{2}-\left(\dfrac{23}{3}+\left(\dfrac{78}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{25}{9}\right)\sqrt{8}\right)\right)\times\left(\left(-3\right)\sqrt{50}+\left(-\dfrac{68}{5}\right)\sqrt{18}+\left(\dfrac{15}{4}\right)\sqrt{4}-\dfrac{54}{5}-\dfrac{74}{7}\right)\\
&=&\left(\dfrac{63}{2}-\left(\dfrac{23}{3}+\left(\dfrac{390}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{50}{9}\right)\sqrt{2}\right)\right)\times\left(\left(-15\right)\sqrt{2}+\left(-\dfrac{204}{5}\right)\sqrt{2}+\dfrac{15}{2}-\dfrac{54}{5}-\dfrac{74}{7}\right)\\
&=&\left(\dfrac{143}{6}+\left(-\dfrac{3860}{63}\right)\sqrt{2}\right)\left(\left(-\dfrac{279}{5}\right)\sqrt{2}-\dfrac{971}{70}\right)\\
&=&\left(-\dfrac{2116799}{4410}\right)\sqrt{2}-\dfrac{138853}{420}+\left(\dfrac{23932}{7}\right)\sqrt{4}\\
&=&\left(-\dfrac{2116799}{4410}\right)\sqrt{2}+\dfrac{2732987}{420}\\
\end{eqnarray*}