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{40}{3}\right)\sqrt{27}\right)-9-\left(\left(\left(-5\right)\sqrt{27}\right)-\left(\left(-\dfrac{41}{5}\right)\sqrt{75}\right)-\dfrac{21}{8}-\left(\left(-\dfrac{9}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)\right)-\left(-\dfrac{77}{6}-\left(\left(0\right)\sqrt{27}\right)\right)$ et $ Y=\left(-\dfrac{11}{6}\right)\sqrt{9}+\left(-\dfrac{19}{2}\right)\sqrt{9}+\left(\dfrac{19}{5}\right)\sqrt{12}+\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{9}\right)-\left(\left(-2\right)\sqrt{27}\right)+\left(\dfrac{26}{5}\right)\sqrt{9}-\dfrac{4}{7}+\left(\dfrac{30}{7}\right)\sqrt{75}+\left(-\dfrac{75}{4}\right)\sqrt{27}$ . 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{40}{3}\right)\sqrt{27}\right)-9-\left(\left(\left(-5\right)\sqrt{27}\right)-\left(\left(-\dfrac{41}{5}\right)\sqrt{75}\right)-\dfrac{21}{8}-\left(\left(-\dfrac{9}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)\right)-\left(-\dfrac{77}{6}-\left(\left(0\right)\sqrt{27}\right)\right)\right)+\left(\left(-\dfrac{11}{6}\right)\sqrt{9}+\left(-\dfrac{19}{2}\right)\sqrt{9}+\left(\dfrac{19}{5}\right)\sqrt{12}+\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{9}\right)-\left(\left(-2\right)\sqrt{27}\right)+\left(\dfrac{26}{5}\right)\sqrt{9}-\dfrac{4}{7}+\left(\dfrac{30}{7}\right)\sqrt{75}+\left(-\dfrac{75}{4}\right)\sqrt{27}\right)\\ &=&\left(\left(\left(-40\right)\sqrt{3}\right)-9-\left(\left(\left(-15\right)\sqrt{3}\right)-\left(\left(-41\right)\sqrt{3}\right)-\dfrac{21}{8}-\left(\left(-9\right)\sqrt{3}\right)-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)\right)-\left(-\dfrac{77}{6}-\left(\left(0\right)\sqrt{3}\right)\right)\right)+\left(-\dfrac{11}{2}-\dfrac{57}{2}+\left(\dfrac{38}{5}\right)\sqrt{3}+\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{38}{5}\right)\sqrt{3}\right)+\dfrac{57}{2}-\left(\left(-6\right)\sqrt{3}\right)+\dfrac{78}{5}-\dfrac{4}{7}+\left(\dfrac{150}{7}\right)\sqrt{3}+\left(-\dfrac{225}{4}\right)\sqrt{3}\right)\\ &=&\left(\left(-40\right)\sqrt{3}\right)-9-\left(\left(\left(-15\right)\sqrt{3}\right)-\left(\left(-41\right)\sqrt{3}\right)-\dfrac{21}{8}-\left(\left(-9\right)\sqrt{3}\right)-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)\right)-\left(-\dfrac{77}{6}-\left(\left(0\right)\sqrt{3}\right)\right)-\dfrac{11}{2}-\dfrac{57}{2}+\left(\dfrac{38}{5}\right)\sqrt{3}+\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{38}{5}\right)\sqrt{3}\right)+\dfrac{57}{2}-\left(\left(-6\right)\sqrt{3}\right)+\dfrac{78}{5}-\dfrac{4}{7}+\left(\dfrac{150}{7}\right)\sqrt{3}+\left(-\dfrac{225}{4}\right)\sqrt{3}\\ &=&\left(-\dfrac{5767}{84}\right)\sqrt{3}+\dfrac{13429}{840}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(-\dfrac{40}{3}\right)\sqrt{27}\right)-9-\left(\left(\left(-5\right)\sqrt{27}\right)-\left(\left(-\dfrac{41}{5}\right)\sqrt{75}\right)-\dfrac{21}{8}-\left(\left(-\dfrac{9}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)\right)-\left(-\dfrac{77}{6}-\left(\left(0\right)\sqrt{27}\right)\right)\right)-\left(\left(-\dfrac{11}{6}\right)\sqrt{9}+\left(-\dfrac{19}{2}\right)\sqrt{9}+\left(\dfrac{19}{5}\right)\sqrt{12}+\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{9}\right)-\left(\left(-2\right)\sqrt{27}\right)+\left(\dfrac{26}{5}\right)\sqrt{9}-\dfrac{4}{7}+\left(\dfrac{30}{7}\right)\sqrt{75}+\left(-\dfrac{75}{4}\right)\sqrt{27}\right)\\ &=&\left(\left(\left(-40\right)\sqrt{3}\right)-9-\left(\left(\left(-15\right)\sqrt{3}\right)-\left(\left(-41\right)\sqrt{3}\right)-\dfrac{21}{8}-\left(\left(-9\right)\sqrt{3}\right)-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)\right)-\left(-\dfrac{77}{6}-\left(\left(0\right)\sqrt{3}\right)\right)\right)-\left(-\dfrac{11}{2}-\dfrac{57}{2}+\left(\dfrac{38}{5}\right)\sqrt{3}+\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{38}{5}\right)\sqrt{3}\right)+\dfrac{57}{2}-\left(\left(-6\right)\sqrt{3}\right)+\dfrac{78}{5}-\dfrac{4}{7}+\left(\dfrac{150}{7}\right)\sqrt{3}+\left(-\dfrac{225}{4}\right)\sqrt{3}\right)\\ &=&\left(\left(-\dfrac{205}{3}\right)\sqrt{3}+\dfrac{155}{24}\right)-\left(\dfrac{667}{70}+\left(-\dfrac{9}{28}\right)\sqrt{3}\right)\\ &=&\left(-\dfrac{205}{3}\right)\sqrt{3}+\dfrac{155}{24}+-\dfrac{667}{70}+\left(\dfrac{9}{28}\right)\sqrt{3}\\ &=&\left(-\dfrac{5713}{84}\right)\sqrt{3}-\dfrac{2579}{840}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(-\dfrac{40}{3}\right)\sqrt{27}\right)-9-\left(\left(\left(-5\right)\sqrt{27}\right)-\left(\left(-\dfrac{41}{5}\right)\sqrt{75}\right)-\dfrac{21}{8}-\left(\left(-\dfrac{9}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)\right)-\left(-\dfrac{77}{6}-\left(\left(0\right)\sqrt{27}\right)\right)\right)\times\left(\left(-\dfrac{11}{6}\right)\sqrt{9}+\left(-\dfrac{19}{2}\right)\sqrt{9}+\left(\dfrac{19}{5}\right)\sqrt{12}+\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{9}\right)-\left(\left(-2\right)\sqrt{27}\right)+\left(\dfrac{26}{5}\right)\sqrt{9}-\dfrac{4}{7}+\left(\dfrac{30}{7}\right)\sqrt{75}+\left(-\dfrac{75}{4}\right)\sqrt{27}\right)\\ &=&\left(\left(\left(-40\right)\sqrt{3}\right)-9-\left(\left(\left(-15\right)\sqrt{3}\right)-\left(\left(-41\right)\sqrt{3}\right)-\dfrac{21}{8}-\left(\left(-9\right)\sqrt{3}\right)-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)\right)-\left(-\dfrac{77}{6}-\left(\left(0\right)\sqrt{3}\right)\right)\right)\times\left(-\dfrac{11}{2}-\dfrac{57}{2}+\left(\dfrac{38}{5}\right)\sqrt{3}+\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{38}{5}\right)\sqrt{3}\right)+\dfrac{57}{2}-\left(\left(-6\right)\sqrt{3}\right)+\dfrac{78}{5}-\dfrac{4}{7}+\left(\dfrac{150}{7}\right)\sqrt{3}+\left(-\dfrac{225}{4}\right)\sqrt{3}\right)\\ &=&\left(\left(-\dfrac{205}{3}\right)\sqrt{3}+\dfrac{155}{24}\right)\left(\dfrac{667}{70}+\left(-\dfrac{9}{28}\right)\sqrt{3}\right)\\ &=&\left(-\dfrac{438947}{672}\right)\sqrt{3}+\left(\dfrac{615}{28}\right)\sqrt{9}+\dfrac{20677}{336}\\ &=&\left(-\dfrac{438947}{672}\right)\sqrt{3}+\dfrac{42817}{336}\\ \end{eqnarray*}