L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{18}{5}\right)\sqrt{45}\) et \( Y=\dfrac{41}{3}-\dfrac{57}{2}+\left(-\dfrac{38}{3}\right)\sqrt{25}+\left(-\dfrac{23}{3}\right)\sqrt{20}+\left(-\dfrac{65}{6}\right)\sqrt{125}+\left(-\dfrac{65}{6}\right)\sqrt{125}\) . 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{18}{5}\right)\sqrt{45}\right)+\left(\dfrac{41}{3}-\dfrac{57}{2}+\left(-\dfrac{38}{3}\right)\sqrt{25}+\left(-\dfrac{23}{3}\right)\sqrt{20}+\left(-\dfrac{65}{6}\right)\sqrt{125}+\left(-\dfrac{65}{6}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{54}{5}\right)\sqrt{5}\right)+\left(\dfrac{41}{3}-\dfrac{57}{2}-\dfrac{190}{3}+\left(-\dfrac{46}{3}\right)\sqrt{5}+\left(-\dfrac{325}{6}\right)\sqrt{5}+\left(-\dfrac{325}{6}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{54}{5}\right)\sqrt{5}+\dfrac{41}{3}-\dfrac{57}{2}-\dfrac{190}{3}+\left(-\dfrac{46}{3}\right)\sqrt{5}+\left(-\dfrac{325}{6}\right)\sqrt{5}+\left(-\dfrac{325}{6}\right)\sqrt{5}\\
&=&\left(-\dfrac{2017}{15}\right)\sqrt{5}-\dfrac{469}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{18}{5}\right)\sqrt{45}\right)-\left(\dfrac{41}{3}-\dfrac{57}{2}+\left(-\dfrac{38}{3}\right)\sqrt{25}+\left(-\dfrac{23}{3}\right)\sqrt{20}+\left(-\dfrac{65}{6}\right)\sqrt{125}+\left(-\dfrac{65}{6}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{54}{5}\right)\sqrt{5}\right)-\left(\dfrac{41}{3}-\dfrac{57}{2}-\dfrac{190}{3}+\left(-\dfrac{46}{3}\right)\sqrt{5}+\left(-\dfrac{325}{6}\right)\sqrt{5}+\left(-\dfrac{325}{6}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{54}{5}\right)\sqrt{5}\right)-\left(-\dfrac{469}{6}+\left(-\dfrac{371}{3}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{54}{5}\right)\sqrt{5}+\dfrac{469}{6}+\left(\dfrac{371}{3}\right)\sqrt{5}\\
&=&\left(\dfrac{1693}{15}\right)\sqrt{5}+\dfrac{469}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{18}{5}\right)\sqrt{45}\right)\times\left(\dfrac{41}{3}-\dfrac{57}{2}+\left(-\dfrac{38}{3}\right)\sqrt{25}+\left(-\dfrac{23}{3}\right)\sqrt{20}+\left(-\dfrac{65}{6}\right)\sqrt{125}+\left(-\dfrac{65}{6}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{54}{5}\right)\sqrt{5}\right)\times\left(\dfrac{41}{3}-\dfrac{57}{2}-\dfrac{190}{3}+\left(-\dfrac{46}{3}\right)\sqrt{5}+\left(-\dfrac{325}{6}\right)\sqrt{5}+\left(-\dfrac{325}{6}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{54}{5}\right)\sqrt{5}\right)\left(-\dfrac{469}{6}+\left(-\dfrac{371}{3}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{4221}{5}\right)\sqrt{5}+\left(\dfrac{6678}{5}\right)\sqrt{25}\\
&=&\left(\dfrac{4221}{5}\right)\sqrt{5}+6678\\
\end{eqnarray*}