L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{9}{2}\right)\sqrt{45}+\left(-6\right)\sqrt{25}+\left(-\dfrac{25}{2}\right)\sqrt{25}+\left(\dfrac{35}{9}\right)\sqrt{125}+\left(\dfrac{71}{7}\right)\sqrt{25}\) et \( Y=\left(-3\right)\sqrt{25}+\left(1\right)\sqrt{125}+\left(-2\right)\sqrt{20}+\left(-\dfrac{48}{7}\right)\sqrt{45}+4\) . 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{9}{2}\right)\sqrt{45}+\left(-6\right)\sqrt{25}+\left(-\dfrac{25}{2}\right)\sqrt{25}+\left(\dfrac{35}{9}\right)\sqrt{125}+\left(\dfrac{71}{7}\right)\sqrt{25}\right)+\left(\left(-3\right)\sqrt{25}+\left(1\right)\sqrt{125}+\left(-2\right)\sqrt{20}+\left(-\dfrac{48}{7}\right)\sqrt{45}+4\right)\\
&=&\left(\left(-\dfrac{27}{2}\right)\sqrt{5}-30-\dfrac{125}{2}+\left(\dfrac{175}{9}\right)\sqrt{5}+\dfrac{355}{7}\right)+\left(-15+\left(5\right)\sqrt{5}+\left(-4\right)\sqrt{5}+\left(-\dfrac{144}{7}\right)\sqrt{5}+4\right)\\
&=&\left(-\dfrac{27}{2}\right)\sqrt{5}-30-\dfrac{125}{2}+\left(\dfrac{175}{9}\right)\sqrt{5}+\dfrac{355}{7}-15+\left(5\right)\sqrt{5}+\left(-4\right)\sqrt{5}+\left(-\dfrac{144}{7}\right)\sqrt{5}+4\\
&=&\left(-\dfrac{1717}{126}\right)\sqrt{5}-\dfrac{739}{14}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{9}{2}\right)\sqrt{45}+\left(-6\right)\sqrt{25}+\left(-\dfrac{25}{2}\right)\sqrt{25}+\left(\dfrac{35}{9}\right)\sqrt{125}+\left(\dfrac{71}{7}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{25}+\left(1\right)\sqrt{125}+\left(-2\right)\sqrt{20}+\left(-\dfrac{48}{7}\right)\sqrt{45}+4\right)\\
&=&\left(\left(-\dfrac{27}{2}\right)\sqrt{5}-30-\dfrac{125}{2}+\left(\dfrac{175}{9}\right)\sqrt{5}+\dfrac{355}{7}\right)-\left(-15+\left(5\right)\sqrt{5}+\left(-4\right)\sqrt{5}+\left(-\dfrac{144}{7}\right)\sqrt{5}+4\right)\\
&=&\left(\left(\dfrac{107}{18}\right)\sqrt{5}-\dfrac{585}{14}\right)-\left(-11+\left(-\dfrac{137}{7}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{107}{18}\right)\sqrt{5}-\dfrac{585}{14}+11+\left(\dfrac{137}{7}\right)\sqrt{5}\\
&=&\left(\dfrac{3215}{126}\right)\sqrt{5}-\dfrac{431}{14}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{9}{2}\right)\sqrt{45}+\left(-6\right)\sqrt{25}+\left(-\dfrac{25}{2}\right)\sqrt{25}+\left(\dfrac{35}{9}\right)\sqrt{125}+\left(\dfrac{71}{7}\right)\sqrt{25}\right)\times\left(\left(-3\right)\sqrt{25}+\left(1\right)\sqrt{125}+\left(-2\right)\sqrt{20}+\left(-\dfrac{48}{7}\right)\sqrt{45}+4\right)\\
&=&\left(\left(-\dfrac{27}{2}\right)\sqrt{5}-30-\dfrac{125}{2}+\left(\dfrac{175}{9}\right)\sqrt{5}+\dfrac{355}{7}\right)\times\left(-15+\left(5\right)\sqrt{5}+\left(-4\right)\sqrt{5}+\left(-\dfrac{144}{7}\right)\sqrt{5}+4\right)\\
&=&\left(\left(\dfrac{107}{18}\right)\sqrt{5}-\dfrac{585}{14}\right)\left(-11+\left(-\dfrac{137}{7}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{331816}{441}\right)\sqrt{5}+\left(-\dfrac{14659}{126}\right)\sqrt{25}+\dfrac{6435}{14}\\
&=&\left(\dfrac{331816}{441}\right)\sqrt{5}-\dfrac{7690}{63}\\
\end{eqnarray*}