L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{68}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{20}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{5}{8}\right)\sqrt{25}\right)\) et \( Y=\left(-\dfrac{7}{5}\right)\sqrt{45}\) . 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{68}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{20}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{5}{8}\right)\sqrt{25}\right)\right)+\left(\left(-\dfrac{7}{5}\right)\sqrt{45}\right)\\
&=&\left(\left(\dfrac{340}{3}-\left(\left(-\dfrac{53}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{5}\right)\right)-\dfrac{25}{8}\right)+\left(\left(-\dfrac{21}{5}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{340}{3}-\left(\left(-\dfrac{53}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{5}\right)\right)-\dfrac{25}{8}+\left(-\dfrac{21}{5}\right)\sqrt{5}\\
&=&\dfrac{2645}{24}+\left(\dfrac{59}{30}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{68}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{20}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{5}{8}\right)\sqrt{25}\right)\right)-\left(\left(-\dfrac{7}{5}\right)\sqrt{45}\right)\\
&=&\left(\left(\dfrac{340}{3}-\left(\left(-\dfrac{53}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{5}\right)\right)-\dfrac{25}{8}\right)-\left(\left(-\dfrac{21}{5}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{2645}{24}+\left(\dfrac{37}{6}\right)\sqrt{5}\right)-\left(\left(-\dfrac{21}{5}\right)\sqrt{5}\right)\\
&=&\dfrac{2645}{24}+\left(\dfrac{37}{6}\right)\sqrt{5}+\left(\dfrac{21}{5}\right)\sqrt{5}\\
&=&\dfrac{2645}{24}+\left(\dfrac{311}{30}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{68}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{20}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{5}{8}\right)\sqrt{25}\right)\right)\times\left(\left(-\dfrac{7}{5}\right)\sqrt{45}\right)\\
&=&\left(\left(\dfrac{340}{3}-\left(\left(-\dfrac{53}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{5}\right)\right)-\dfrac{25}{8}\right)\times\left(\left(-\dfrac{21}{5}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{2645}{24}+\left(\dfrac{37}{6}\right)\sqrt{5}\right)\left(\left(-\dfrac{21}{5}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{3703}{8}\right)\sqrt{5}+\left(-\dfrac{259}{10}\right)\sqrt{25}\\
&=&\left(-\dfrac{3703}{8}\right)\sqrt{5}-\dfrac{259}{2}\\
\end{eqnarray*}