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=\dfrac{4}{7}-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{75}\right)-\left(\left(8\right)\sqrt{27}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{12}\right)$ et $ Y=\left(\left(1\right)\sqrt{75}+\left(-\dfrac{13}{3}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{27}\right)+\dfrac{79}{6}-\left(\left(\dfrac{9}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{38}{3}\right)\sqrt{12}\right)\right)-\left(\left(\left(-2\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{28}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{27}\right)-\dfrac{37}{7}-\left(\left(3\right)\sqrt{9}\right)\right)$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(\dfrac{4}{7}-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{75}\right)-\left(\left(8\right)\sqrt{27}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{12}\right)\right)+\left(\left(\left(1\right)\sqrt{75}+\left(-\dfrac{13}{3}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{27}\right)+\dfrac{79}{6}-\left(\left(\dfrac{9}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{38}{3}\right)\sqrt{12}\right)\right)-\left(\left(\left(-2\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{28}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{27}\right)-\dfrac{37}{7}-\left(\left(3\right)\sqrt{9}\right)\right)\right)\\ &=&\left(\dfrac{4}{7}-\left(\left(-25\right)\sqrt{3}\right)-\left(\left(\dfrac{140}{3}\right)\sqrt{3}\right)-\left(\left(24\right)\sqrt{3}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{3}\right)\right)+\left(\left(\left(5\right)\sqrt{3}-13\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{3}\right)+\dfrac{79}{6}-\dfrac{27}{4}-\left(\left(-\dfrac{76}{3}\right)\sqrt{3}\right)\right)-\left(\left(\left(-4\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)+31-\left(\left(\dfrac{93}{4}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)-\left(\dfrac{28}{3}-\left(\left(\dfrac{72}{5}\right)\sqrt{3}\right)-\dfrac{37}{7}-9\right)\right)\\ &=&\dfrac{4}{7}-\left(\left(-25\right)\sqrt{3}\right)-\left(\left(\dfrac{140}{3}\right)\sqrt{3}\right)-\left(\left(24\right)\sqrt{3}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{3}\right)+\left(\left(5\right)\sqrt{3}-13\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{3}\right)+\dfrac{79}{6}-\dfrac{27}{4}-\left(\left(-\dfrac{76}{3}\right)\sqrt{3}\right)\right)-\left(\left(\left(-4\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)+31-\left(\left(\dfrac{93}{4}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)-\left(\dfrac{28}{3}-\left(\left(\dfrac{72}{5}\right)\sqrt{3}\right)-\dfrac{37}{7}-9\right)\\ &=&-\dfrac{1257}{28}+\left(-\dfrac{373}{280}\right)\sqrt{3}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\dfrac{4}{7}-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{75}\right)-\left(\left(8\right)\sqrt{27}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{12}\right)\right)-\left(\left(\left(1\right)\sqrt{75}+\left(-\dfrac{13}{3}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{27}\right)+\dfrac{79}{6}-\left(\left(\dfrac{9}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{38}{3}\right)\sqrt{12}\right)\right)-\left(\left(\left(-2\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{28}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{27}\right)-\dfrac{37}{7}-\left(\left(3\right)\sqrt{9}\right)\right)\right)\\ &=&\left(\dfrac{4}{7}-\left(\left(-25\right)\sqrt{3}\right)-\left(\left(\dfrac{140}{3}\right)\sqrt{3}\right)-\left(\left(24\right)\sqrt{3}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{3}\right)\right)-\left(\left(\left(5\right)\sqrt{3}-13\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{3}\right)+\dfrac{79}{6}-\dfrac{27}{4}-\left(\left(-\dfrac{76}{3}\right)\sqrt{3}\right)\right)-\left(\left(\left(-4\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)+31-\left(\left(\dfrac{93}{4}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)-\left(\dfrac{28}{3}-\left(\left(\dfrac{72}{5}\right)\sqrt{3}\right)-\dfrac{37}{7}-9\right)\right)\\ &=&\left(\dfrac{4}{7}+\left(-\dfrac{157}{6}\right)\sqrt{3}\right)-\left(\left(\dfrac{20861}{840}\right)\sqrt{3}-\dfrac{1273}{28}\right)\\ &=&\dfrac{4}{7}+\left(-\dfrac{157}{6}\right)\sqrt{3}+\left(-\dfrac{20861}{840}\right)\sqrt{3}+\dfrac{1273}{28}\\ &=&\dfrac{1289}{28}+\left(-\dfrac{42841}{840}\right)\sqrt{3}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\dfrac{4}{7}-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{75}\right)-\left(\left(8\right)\sqrt{27}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{12}\right)\right)\times\left(\left(\left(1\right)\sqrt{75}+\left(-\dfrac{13}{3}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{27}\right)+\dfrac{79}{6}-\left(\left(\dfrac{9}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{38}{3}\right)\sqrt{12}\right)\right)-\left(\left(\left(-2\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{10}{7}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{28}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{27}\right)-\dfrac{37}{7}-\left(\left(3\right)\sqrt{9}\right)\right)\right)\\ &=&\left(\dfrac{4}{7}-\left(\left(-25\right)\sqrt{3}\right)-\left(\left(\dfrac{140}{3}\right)\sqrt{3}\right)-\left(\left(24\right)\sqrt{3}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{3}\right)\right)\times\left(\left(\left(5\right)\sqrt{3}-13\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{3}\right)+\dfrac{79}{6}-\dfrac{27}{4}-\left(\left(-\dfrac{76}{3}\right)\sqrt{3}\right)\right)-\left(\left(\left(-4\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)+31-\left(\left(\dfrac{93}{4}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)-\left(\dfrac{28}{3}-\left(\left(\dfrac{72}{5}\right)\sqrt{3}\right)-\dfrac{37}{7}-9\right)\right)\\ &=&\left(\dfrac{4}{7}+\left(-\dfrac{157}{6}\right)\sqrt{3}\right)\left(\left(\dfrac{20861}{840}\right)\sqrt{3}-\dfrac{1273}{28}\right)\\ &=&\left(\dfrac{7078579}{5880}\right)\sqrt{3}-\dfrac{1273}{49}+\left(-\dfrac{3275177}{5040}\right)\sqrt{9}\\ &=&\left(\dfrac{7078579}{5880}\right)\sqrt{3}-\dfrac{23231759}{11760}\\ \end{eqnarray*}