L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{54}{5}\right)\sqrt{45}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{45}\right)-9\right)-\left(\left(\left(-\dfrac{75}{8}\right)\sqrt{25}\right)+\dfrac{12}{7}-\left(\left(-\dfrac{4}{3}\right)\sqrt{45}\right)\right)\) et \( Y=\left(\dfrac{15}{2}\right)\sqrt{20}+\left(-\dfrac{13}{6}\right)\sqrt{45}\) . 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{54}{5}\right)\sqrt{45}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{45}\right)-9\right)-\left(\left(\left(-\dfrac{75}{8}\right)\sqrt{25}\right)+\dfrac{12}{7}-\left(\left(-\dfrac{4}{3}\right)\sqrt{45}\right)\right)\right)+\left(\left(\dfrac{15}{2}\right)\sqrt{20}+\left(-\dfrac{13}{6}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\left(\dfrac{162}{5}\right)\sqrt{5}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-9\right)-\left(-\dfrac{375}{8}+\dfrac{12}{7}-\left(\left(-4\right)\sqrt{5}\right)\right)\right)+\left(\left(15\right)\sqrt{5}+\left(-\dfrac{13}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\left(\dfrac{162}{5}\right)\sqrt{5}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-9\right)-\left(-\dfrac{375}{8}+\dfrac{12}{7}-\left(\left(-4\right)\sqrt{5}\right)\right)+\left(15\right)\sqrt{5}+\left(-\dfrac{13}{2}\right)\sqrt{5}\\
&=&\left(\dfrac{297}{5}\right)\sqrt{5}+\dfrac{2025}{56}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{54}{5}\right)\sqrt{45}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{45}\right)-9\right)-\left(\left(\left(-\dfrac{75}{8}\right)\sqrt{25}\right)+\dfrac{12}{7}-\left(\left(-\dfrac{4}{3}\right)\sqrt{45}\right)\right)\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{20}+\left(-\dfrac{13}{6}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\left(\dfrac{162}{5}\right)\sqrt{5}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-9\right)-\left(-\dfrac{375}{8}+\dfrac{12}{7}-\left(\left(-4\right)\sqrt{5}\right)\right)\right)-\left(\left(15\right)\sqrt{5}+\left(-\dfrac{13}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{509}{10}\right)\sqrt{5}+\dfrac{2025}{56}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{509}{10}\right)\sqrt{5}+\dfrac{2025}{56}+\left(-\dfrac{17}{2}\right)\sqrt{5}\\
&=&\left(\dfrac{212}{5}\right)\sqrt{5}+\dfrac{2025}{56}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{54}{5}\right)\sqrt{45}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{45}\right)-9\right)-\left(\left(\left(-\dfrac{75}{8}\right)\sqrt{25}\right)+\dfrac{12}{7}-\left(\left(-\dfrac{4}{3}\right)\sqrt{45}\right)\right)\right)\times\left(\left(\dfrac{15}{2}\right)\sqrt{20}+\left(-\dfrac{13}{6}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\left(\dfrac{162}{5}\right)\sqrt{5}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-9\right)-\left(-\dfrac{375}{8}+\dfrac{12}{7}-\left(\left(-4\right)\sqrt{5}\right)\right)\right)\times\left(\left(15\right)\sqrt{5}+\left(-\dfrac{13}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{509}{10}\right)\sqrt{5}+\dfrac{2025}{56}\right)\left(\left(\dfrac{17}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{8653}{20}\right)\sqrt{25}+\left(\dfrac{34425}{112}\right)\sqrt{5}\\
&=&\dfrac{8653}{4}+\left(\dfrac{34425}{112}\right)\sqrt{5}\\
\end{eqnarray*}