L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{5}{8}\right)\sqrt{8}+\left(\dfrac{22}{5}\right)\sqrt{18}+3+\left(\dfrac{43}{9}\right)\sqrt{50}+\left(\dfrac{44}{7}\right)\sqrt{18}+\left(-\dfrac{28}{5}\right)\sqrt{50}+\left(-6\right)\sqrt{4}\) et \( Y=\dfrac{76}{7}\) . 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{5}{8}\right)\sqrt{8}+\left(\dfrac{22}{5}\right)\sqrt{18}+3+\left(\dfrac{43}{9}\right)\sqrt{50}+\left(\dfrac{44}{7}\right)\sqrt{18}+\left(-\dfrac{28}{5}\right)\sqrt{50}+\left(-6\right)\sqrt{4}\right)+\left(\dfrac{76}{7}\right)\\
&=&\left(\left(\dfrac{5}{4}\right)\sqrt{2}+\left(\dfrac{66}{5}\right)\sqrt{2}+3+\left(\dfrac{215}{9}\right)\sqrt{2}+\left(\dfrac{132}{7}\right)\sqrt{2}+\left(-28\right)\sqrt{2}-12\right)+\left(\dfrac{76}{7}\right)\\
&=&\left(\dfrac{5}{4}\right)\sqrt{2}+\left(\dfrac{66}{5}\right)\sqrt{2}+3+\left(\dfrac{215}{9}\right)\sqrt{2}+\left(\dfrac{132}{7}\right)\sqrt{2}+\left(-28\right)\sqrt{2}-12+\dfrac{76}{7}\\
&=&\left(\dfrac{36787}{1260}\right)\sqrt{2}+\dfrac{13}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{5}{8}\right)\sqrt{8}+\left(\dfrac{22}{5}\right)\sqrt{18}+3+\left(\dfrac{43}{9}\right)\sqrt{50}+\left(\dfrac{44}{7}\right)\sqrt{18}+\left(-\dfrac{28}{5}\right)\sqrt{50}+\left(-6\right)\sqrt{4}\right)-\left(\dfrac{76}{7}\right)\\
&=&\left(\left(\dfrac{5}{4}\right)\sqrt{2}+\left(\dfrac{66}{5}\right)\sqrt{2}+3+\left(\dfrac{215}{9}\right)\sqrt{2}+\left(\dfrac{132}{7}\right)\sqrt{2}+\left(-28\right)\sqrt{2}-12\right)-\left(\dfrac{76}{7}\right)\\
&=&\left(\left(\dfrac{36787}{1260}\right)\sqrt{2}-9\right)-\left(\dfrac{76}{7}\right)\\
&=&\left(\dfrac{36787}{1260}\right)\sqrt{2}-9+-\dfrac{76}{7}\\
&=&\left(\dfrac{36787}{1260}\right)\sqrt{2}-\dfrac{139}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{5}{8}\right)\sqrt{8}+\left(\dfrac{22}{5}\right)\sqrt{18}+3+\left(\dfrac{43}{9}\right)\sqrt{50}+\left(\dfrac{44}{7}\right)\sqrt{18}+\left(-\dfrac{28}{5}\right)\sqrt{50}+\left(-6\right)\sqrt{4}\right)\times\left(\dfrac{76}{7}\right)\\
&=&\left(\left(\dfrac{5}{4}\right)\sqrt{2}+\left(\dfrac{66}{5}\right)\sqrt{2}+3+\left(\dfrac{215}{9}\right)\sqrt{2}+\left(\dfrac{132}{7}\right)\sqrt{2}+\left(-28\right)\sqrt{2}-12\right)\times\left(\dfrac{76}{7}\right)\\
&=&\left(\left(\dfrac{36787}{1260}\right)\sqrt{2}-9\right)\left(\dfrac{76}{7}\right)\\
&=&\left(\dfrac{698953}{2205}\right)\sqrt{2}-\dfrac{684}{7}\\
&=&\left(\dfrac{698953}{2205}\right)\sqrt{2}-\dfrac{684}{7}\\
\end{eqnarray*}