Ataraxy

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Exercice

L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.

Exercice

Soit $ X=\left(\left(-\dfrac{10}{3}\right)\sqrt{75}\right)-0-\left(\left(-\dfrac{55}{2}\right)\sqrt{75}\right)-\left(\left(\dfrac{55}{8}\right)\sqrt{9}\right)-\left(\left(-\dfrac{7}{5}\right)\sqrt{27}\right)+\left(\dfrac{72}{5}\right)\sqrt{27}+\left(-\dfrac{11}{3}\right)\sqrt{27}+\left(-\dfrac{5}{2}\right)\sqrt{9}+\dfrac{75}{4}+\left(\dfrac{49}{8}\right)\sqrt{75}$ et $ Y=\left(-\dfrac{47}{7}\right)\sqrt{9}+\left(\left(\dfrac{26}{3}\right)\sqrt{12}\right)-\left(\left(\dfrac{75}{8}\right)\sqrt{12}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{27}\right)+\left(\left(\dfrac{26}{3}\right)\sqrt{12}\right)+3-\left(\left(\dfrac{10}{9}\right)\sqrt{75}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{75}\right)+\left(-\dfrac{13}{8}\right)\sqrt{12}$ . 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(-\dfrac{10}{3}\right)\sqrt{75}\right)-0-\left(\left(-\dfrac{55}{2}\right)\sqrt{75}\right)-\left(\left(\dfrac{55}{8}\right)\sqrt{9}\right)-\left(\left(-\dfrac{7}{5}\right)\sqrt{27}\right)+\left(\dfrac{72}{5}\right)\sqrt{27}+\left(-\dfrac{11}{3}\right)\sqrt{27}+\left(-\dfrac{5}{2}\right)\sqrt{9}+\dfrac{75}{4}+\left(\dfrac{49}{8}\right)\sqrt{75}\right)+\left(\left(-\dfrac{47}{7}\right)\sqrt{9}+\left(\left(\dfrac{26}{3}\right)\sqrt{12}\right)-\left(\left(\dfrac{75}{8}\right)\sqrt{12}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{27}\right)+\left(\left(\dfrac{26}{3}\right)\sqrt{12}\right)+3-\left(\left(\dfrac{10}{9}\right)\sqrt{75}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{75}\right)+\left(-\dfrac{13}{8}\right)\sqrt{12}\right)\\ &=&\left(\left(\left(-\dfrac{50}{3}\right)\sqrt{3}\right)-0-\left(\left(-\dfrac{275}{2}\right)\sqrt{3}\right)-\dfrac{165}{8}-\left(\left(-\dfrac{21}{5}\right)\sqrt{3}\right)+\left(\dfrac{216}{5}\right)\sqrt{3}+\left(-11\right)\sqrt{3}-\dfrac{15}{2}+\dfrac{75}{4}+\left(\dfrac{245}{8}\right)\sqrt{3}\right)+\left(-\dfrac{141}{7}+\left(\left(\dfrac{52}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{75}{4}\right)\sqrt{3}\right)+\dfrac{99}{2}-\left(\left(-15\right)\sqrt{3}\right)+\left(\left(\dfrac{52}{3}\right)\sqrt{3}\right)+3-\left(\left(\dfrac{50}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{255}{4}\right)\sqrt{3}\right)+\left(-\dfrac{13}{4}\right)\sqrt{3}\right)\\ &=&\left(\left(-\dfrac{50}{3}\right)\sqrt{3}\right)-0-\left(\left(-\dfrac{275}{2}\right)\sqrt{3}\right)-\dfrac{165}{8}-\left(\left(-\dfrac{21}{5}\right)\sqrt{3}\right)+\left(\dfrac{216}{5}\right)\sqrt{3}+\left(-11\right)\sqrt{3}-\dfrac{15}{2}+\dfrac{75}{4}+\left(\dfrac{245}{8}\right)\sqrt{3}-\dfrac{141}{7}+\left(\left(\dfrac{52}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{75}{4}\right)\sqrt{3}\right)+\dfrac{99}{2}-\left(\left(-15\right)\sqrt{3}\right)+\left(\left(\dfrac{52}{3}\right)\sqrt{3}\right)+3-\left(\left(\dfrac{50}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{255}{4}\right)\sqrt{3}\right)+\left(-\dfrac{13}{4}\right)\sqrt{3}\\ &=&\left(\dfrac{98539}{360}\right)\sqrt{3}+\dfrac{1287}{56}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{75}\right)-0-\left(\left(-\dfrac{55}{2}\right)\sqrt{75}\right)-\left(\left(\dfrac{55}{8}\right)\sqrt{9}\right)-\left(\left(-\dfrac{7}{5}\right)\sqrt{27}\right)+\left(\dfrac{72}{5}\right)\sqrt{27}+\left(-\dfrac{11}{3}\right)\sqrt{27}+\left(-\dfrac{5}{2}\right)\sqrt{9}+\dfrac{75}{4}+\left(\dfrac{49}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{47}{7}\right)\sqrt{9}+\left(\left(\dfrac{26}{3}\right)\sqrt{12}\right)-\left(\left(\dfrac{75}{8}\right)\sqrt{12}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{27}\right)+\left(\left(\dfrac{26}{3}\right)\sqrt{12}\right)+3-\left(\left(\dfrac{10}{9}\right)\sqrt{75}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{75}\right)+\left(-\dfrac{13}{8}\right)\sqrt{12}\right)\\ &=&\left(\left(\left(-\dfrac{50}{3}\right)\sqrt{3}\right)-0-\left(\left(-\dfrac{275}{2}\right)\sqrt{3}\right)-\dfrac{165}{8}-\left(\left(-\dfrac{21}{5}\right)\sqrt{3}\right)+\left(\dfrac{216}{5}\right)\sqrt{3}+\left(-11\right)\sqrt{3}-\dfrac{15}{2}+\dfrac{75}{4}+\left(\dfrac{245}{8}\right)\sqrt{3}\right)-\left(-\dfrac{141}{7}+\left(\left(\dfrac{52}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{75}{4}\right)\sqrt{3}\right)+\dfrac{99}{2}-\left(\left(-15\right)\sqrt{3}\right)+\left(\left(\dfrac{52}{3}\right)\sqrt{3}\right)+3-\left(\left(\dfrac{50}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{255}{4}\right)\sqrt{3}\right)+\left(-\dfrac{13}{4}\right)\sqrt{3}\right)\\ &=&\left(\left(\dfrac{22543}{120}\right)\sqrt{3}-\dfrac{75}{8}\right)-\left(\dfrac{453}{14}+\left(\dfrac{3091}{36}\right)\sqrt{3}\right)\\ &=&\left(\dfrac{22543}{120}\right)\sqrt{3}-\dfrac{75}{8}+-\dfrac{453}{14}+\left(-\dfrac{3091}{36}\right)\sqrt{3}\\ &=&\left(\dfrac{36719}{360}\right)\sqrt{3}-\dfrac{2337}{56}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{75}\right)-0-\left(\left(-\dfrac{55}{2}\right)\sqrt{75}\right)-\left(\left(\dfrac{55}{8}\right)\sqrt{9}\right)-\left(\left(-\dfrac{7}{5}\right)\sqrt{27}\right)+\left(\dfrac{72}{5}\right)\sqrt{27}+\left(-\dfrac{11}{3}\right)\sqrt{27}+\left(-\dfrac{5}{2}\right)\sqrt{9}+\dfrac{75}{4}+\left(\dfrac{49}{8}\right)\sqrt{75}\right)\times\left(\left(-\dfrac{47}{7}\right)\sqrt{9}+\left(\left(\dfrac{26}{3}\right)\sqrt{12}\right)-\left(\left(\dfrac{75}{8}\right)\sqrt{12}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{27}\right)+\left(\left(\dfrac{26}{3}\right)\sqrt{12}\right)+3-\left(\left(\dfrac{10}{9}\right)\sqrt{75}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{75}\right)+\left(-\dfrac{13}{8}\right)\sqrt{12}\right)\\ &=&\left(\left(\left(-\dfrac{50}{3}\right)\sqrt{3}\right)-0-\left(\left(-\dfrac{275}{2}\right)\sqrt{3}\right)-\dfrac{165}{8}-\left(\left(-\dfrac{21}{5}\right)\sqrt{3}\right)+\left(\dfrac{216}{5}\right)\sqrt{3}+\left(-11\right)\sqrt{3}-\dfrac{15}{2}+\dfrac{75}{4}+\left(\dfrac{245}{8}\right)\sqrt{3}\right)\times\left(-\dfrac{141}{7}+\left(\left(\dfrac{52}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{75}{4}\right)\sqrt{3}\right)+\dfrac{99}{2}-\left(\left(-15\right)\sqrt{3}\right)+\left(\left(\dfrac{52}{3}\right)\sqrt{3}\right)+3-\left(\left(\dfrac{50}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{255}{4}\right)\sqrt{3}\right)+\left(-\dfrac{13}{4}\right)\sqrt{3}\right)\\ &=&\left(\left(\dfrac{22543}{120}\right)\sqrt{3}-\dfrac{75}{8}\right)\left(\dfrac{453}{14}+\left(\dfrac{3091}{36}\right)\sqrt{3}\right)\\ &=&\left(\dfrac{17719333}{3360}\right)\sqrt{3}+\left(\dfrac{69680413}{4320}\right)\sqrt{9}-\dfrac{33975}{112}\\ &=&\left(\dfrac{17719333}{3360}\right)\sqrt{3}+\dfrac{7755282256}{161280}\\ \end{eqnarray*}