L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{38}{3}\right)\sqrt{45}+\left(-\dfrac{2}{9}\right)\sqrt{125}+\left(-9\right)\sqrt{45}+\dfrac{41}{4}-8+\left(\dfrac{77}{9}\right)\sqrt{25}\) et \( Y=-7-\left(\left(\dfrac{11}{6}\right)\sqrt{20}\right)-4+\left(\left(\dfrac{38}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{20}\right)+6-\left(\left(-\dfrac{72}{5}\right)\sqrt{20}\right)+\left(\left(-8\right)\sqrt{125}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{45}\right)-\dfrac{73}{7}+\left(-\dfrac{38}{9}\right)\sqrt{45}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{38}{3}\right)\sqrt{45}+\left(-\dfrac{2}{9}\right)\sqrt{125}+\left(-9\right)\sqrt{45}+\dfrac{41}{4}-8+\left(\dfrac{77}{9}\right)\sqrt{25}\right)+\left(-7-\left(\left(\dfrac{11}{6}\right)\sqrt{20}\right)-4+\left(\left(\dfrac{38}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{20}\right)+6-\left(\left(-\dfrac{72}{5}\right)\sqrt{20}\right)+\left(\left(-8\right)\sqrt{125}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{45}\right)-\dfrac{73}{7}+\left(-\dfrac{38}{9}\right)\sqrt{45}\right)\\
&=&\left(\left(-38\right)\sqrt{5}+\left(-\dfrac{10}{9}\right)\sqrt{5}+\left(-27\right)\sqrt{5}+\dfrac{41}{4}-8+\dfrac{385}{9}\right)+\left(-7-\left(\left(\dfrac{11}{3}\right)\sqrt{5}\right)-4+\left(\left(38\right)\sqrt{5}\right)-\left(\left(-\dfrac{75}{4}\right)\sqrt{5}\right)+6-\left(\left(-\dfrac{144}{5}\right)\sqrt{5}\right)+\left(\left(-40\right)\sqrt{5}\right)-\left(\left(\dfrac{53}{3}\right)\sqrt{5}\right)-\dfrac{73}{7}+\left(-\dfrac{38}{3}\right)\sqrt{5}\right)\\
&=&\left(-38\right)\sqrt{5}+\left(-\dfrac{10}{9}\right)\sqrt{5}+\left(-27\right)\sqrt{5}+\dfrac{41}{4}-8+\dfrac{385}{9}-7-\left(\left(\dfrac{11}{3}\right)\sqrt{5}\right)-4+\left(\left(38\right)\sqrt{5}\right)-\left(\left(-\dfrac{75}{4}\right)\sqrt{5}\right)+6-\left(\left(-\dfrac{144}{5}\right)\sqrt{5}\right)+\left(\left(-40\right)\sqrt{5}\right)-\left(\left(\dfrac{53}{3}\right)\sqrt{5}\right)-\dfrac{73}{7}+\left(-\dfrac{38}{3}\right)\sqrt{5}\\
&=&\left(-\dfrac{9821}{180}\right)\sqrt{5}+\dfrac{7459}{252}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{38}{3}\right)\sqrt{45}+\left(-\dfrac{2}{9}\right)\sqrt{125}+\left(-9\right)\sqrt{45}+\dfrac{41}{4}-8+\left(\dfrac{77}{9}\right)\sqrt{25}\right)-\left(-7-\left(\left(\dfrac{11}{6}\right)\sqrt{20}\right)-4+\left(\left(\dfrac{38}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{20}\right)+6-\left(\left(-\dfrac{72}{5}\right)\sqrt{20}\right)+\left(\left(-8\right)\sqrt{125}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{45}\right)-\dfrac{73}{7}+\left(-\dfrac{38}{9}\right)\sqrt{45}\right)\\
&=&\left(\left(-38\right)\sqrt{5}+\left(-\dfrac{10}{9}\right)\sqrt{5}+\left(-27\right)\sqrt{5}+\dfrac{41}{4}-8+\dfrac{385}{9}\right)-\left(-7-\left(\left(\dfrac{11}{3}\right)\sqrt{5}\right)-4+\left(\left(38\right)\sqrt{5}\right)-\left(\left(-\dfrac{75}{4}\right)\sqrt{5}\right)+6-\left(\left(-\dfrac{144}{5}\right)\sqrt{5}\right)+\left(\left(-40\right)\sqrt{5}\right)-\left(\left(\dfrac{53}{3}\right)\sqrt{5}\right)-\dfrac{73}{7}+\left(-\dfrac{38}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{595}{9}\right)\sqrt{5}+\dfrac{1621}{36}\right)-\left(-\dfrac{108}{7}+\left(\dfrac{231}{20}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{595}{9}\right)\sqrt{5}+\dfrac{1621}{36}+\dfrac{108}{7}+\left(-\dfrac{231}{20}\right)\sqrt{5}\\
&=&\left(-\dfrac{13979}{180}\right)\sqrt{5}+\dfrac{15235}{252}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{38}{3}\right)\sqrt{45}+\left(-\dfrac{2}{9}\right)\sqrt{125}+\left(-9\right)\sqrt{45}+\dfrac{41}{4}-8+\left(\dfrac{77}{9}\right)\sqrt{25}\right)\times\left(-7-\left(\left(\dfrac{11}{6}\right)\sqrt{20}\right)-4+\left(\left(\dfrac{38}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{20}\right)+6-\left(\left(-\dfrac{72}{5}\right)\sqrt{20}\right)+\left(\left(-8\right)\sqrt{125}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{45}\right)-\dfrac{73}{7}+\left(-\dfrac{38}{9}\right)\sqrt{45}\right)\\
&=&\left(\left(-38\right)\sqrt{5}+\left(-\dfrac{10}{9}\right)\sqrt{5}+\left(-27\right)\sqrt{5}+\dfrac{41}{4}-8+\dfrac{385}{9}\right)\times\left(-7-\left(\left(\dfrac{11}{3}\right)\sqrt{5}\right)-4+\left(\left(38\right)\sqrt{5}\right)-\left(\left(-\dfrac{75}{4}\right)\sqrt{5}\right)+6-\left(\left(-\dfrac{144}{5}\right)\sqrt{5}\right)+\left(\left(-40\right)\sqrt{5}\right)-\left(\left(\dfrac{53}{3}\right)\sqrt{5}\right)-\dfrac{73}{7}+\left(-\dfrac{38}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{595}{9}\right)\sqrt{5}+\dfrac{1621}{36}\right)\left(-\dfrac{108}{7}+\left(\dfrac{231}{20}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{369617}{240}\right)\sqrt{5}+\left(-\dfrac{9163}{12}\right)\sqrt{25}-\dfrac{4863}{7}\\
&=&\left(\dfrac{369617}{240}\right)\sqrt{5}-\dfrac{379061}{84}\\
\end{eqnarray*}