L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{15}{8}\right)\sqrt{20}+\left(-\dfrac{7}{2}\right)\sqrt{25}+\left(-\dfrac{71}{3}\right)\sqrt{25}+\left(7\right)\sqrt{20}\) et \( Y=\left(\dfrac{17}{3}\right)\sqrt{125}+\left(-\dfrac{26}{3}\right)\sqrt{25}+\left(3\right)\sqrt{20}+\left(\dfrac{76}{5}\right)\sqrt{20}+\left(\dfrac{17}{3}\right)\sqrt{125}\) . 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{15}{8}\right)\sqrt{20}+\left(-\dfrac{7}{2}\right)\sqrt{25}+\left(-\dfrac{71}{3}\right)\sqrt{25}+\left(7\right)\sqrt{20}\right)+\left(\left(\dfrac{17}{3}\right)\sqrt{125}+\left(-\dfrac{26}{3}\right)\sqrt{25}+\left(3\right)\sqrt{20}+\left(\dfrac{76}{5}\right)\sqrt{20}+\left(\dfrac{17}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{15}{4}\right)\sqrt{5}-\dfrac{35}{2}-\dfrac{355}{3}+\left(14\right)\sqrt{5}\right)+\left(\left(\dfrac{85}{3}\right)\sqrt{5}-\dfrac{130}{3}+\left(6\right)\sqrt{5}+\left(\dfrac{152}{5}\right)\sqrt{5}+\left(\dfrac{85}{3}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{15}{4}\right)\sqrt{5}-\dfrac{35}{2}-\dfrac{355}{3}+\left(14\right)\sqrt{5}+\left(\dfrac{85}{3}\right)\sqrt{5}-\dfrac{130}{3}+\left(6\right)\sqrt{5}+\left(\dfrac{152}{5}\right)\sqrt{5}+\left(\dfrac{85}{3}\right)\sqrt{5}\\
&=&\left(\dfrac{6199}{60}\right)\sqrt{5}-\dfrac{1075}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{15}{8}\right)\sqrt{20}+\left(-\dfrac{7}{2}\right)\sqrt{25}+\left(-\dfrac{71}{3}\right)\sqrt{25}+\left(7\right)\sqrt{20}\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{125}+\left(-\dfrac{26}{3}\right)\sqrt{25}+\left(3\right)\sqrt{20}+\left(\dfrac{76}{5}\right)\sqrt{20}+\left(\dfrac{17}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{15}{4}\right)\sqrt{5}-\dfrac{35}{2}-\dfrac{355}{3}+\left(14\right)\sqrt{5}\right)-\left(\left(\dfrac{85}{3}\right)\sqrt{5}-\dfrac{130}{3}+\left(6\right)\sqrt{5}+\left(\dfrac{152}{5}\right)\sqrt{5}+\left(\dfrac{85}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{41}{4}\right)\sqrt{5}-\dfrac{815}{6}\right)-\left(\left(\dfrac{1396}{15}\right)\sqrt{5}-\dfrac{130}{3}\right)\\
&=&\left(\dfrac{41}{4}\right)\sqrt{5}-\dfrac{815}{6}+\left(-\dfrac{1396}{15}\right)\sqrt{5}+\dfrac{130}{3}\\
&=&\left(-\dfrac{4969}{60}\right)\sqrt{5}-\dfrac{185}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{15}{8}\right)\sqrt{20}+\left(-\dfrac{7}{2}\right)\sqrt{25}+\left(-\dfrac{71}{3}\right)\sqrt{25}+\left(7\right)\sqrt{20}\right)\times\left(\left(\dfrac{17}{3}\right)\sqrt{125}+\left(-\dfrac{26}{3}\right)\sqrt{25}+\left(3\right)\sqrt{20}+\left(\dfrac{76}{5}\right)\sqrt{20}+\left(\dfrac{17}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{15}{4}\right)\sqrt{5}-\dfrac{35}{2}-\dfrac{355}{3}+\left(14\right)\sqrt{5}\right)\times\left(\left(\dfrac{85}{3}\right)\sqrt{5}-\dfrac{130}{3}+\left(6\right)\sqrt{5}+\left(\dfrac{152}{5}\right)\sqrt{5}+\left(\dfrac{85}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{41}{4}\right)\sqrt{5}-\dfrac{815}{6}\right)\left(\left(\dfrac{1396}{15}\right)\sqrt{5}-\dfrac{130}{3}\right)\\
&=&\left(\dfrac{14309}{15}\right)\sqrt{25}+\left(-\dfrac{235543}{18}\right)\sqrt{5}+\dfrac{52975}{9}\\
&=&\dfrac{95902}{9}+\left(-\dfrac{235543}{18}\right)\sqrt{5}\\
\end{eqnarray*}