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{77}{4}\right)\sqrt{9}+\left(\dfrac{23}{3}\right)\sqrt{12}+\left(-\dfrac{53}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{9}+\left(-8\right)\sqrt{27}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{9}+\left(\dfrac{51}{2}\right)\sqrt{75}+\left(\dfrac{30}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{12}+\left(-\dfrac{51}{2}\right)\sqrt{9}+\left(-\dfrac{5}{2}\right)\sqrt{75}+\left(-\dfrac{13}{9}\right)\sqrt{12}\right)$ et $ Y=\left(\left(-\dfrac{53}{5}\right)\sqrt{9}\right)-\left(\left(-6\right)\sqrt{75}+\left(3\right)\sqrt{27}+\left(-\dfrac{53}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{27}+\left(\dfrac{52}{9}\right)\sqrt{75}\right)$ . 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{77}{4}\right)\sqrt{9}+\left(\dfrac{23}{3}\right)\sqrt{12}+\left(-\dfrac{53}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{9}+\left(-8\right)\sqrt{27}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{9}+\left(\dfrac{51}{2}\right)\sqrt{75}+\left(\dfrac{30}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{12}+\left(-\dfrac{51}{2}\right)\sqrt{9}+\left(-\dfrac{5}{2}\right)\sqrt{75}+\left(-\dfrac{13}{9}\right)\sqrt{12}\right)\right)+\left(\left(\left(-\dfrac{53}{5}\right)\sqrt{9}\right)-\left(\left(-6\right)\sqrt{75}+\left(3\right)\sqrt{27}+\left(-\dfrac{53}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{27}+\left(\dfrac{52}{9}\right)\sqrt{75}\right)\right)\\ &=&\left(\left(\dfrac{231}{4}+\left(\dfrac{46}{3}\right)\sqrt{3}-\dfrac{159}{2}\right)-\left(\left(\dfrac{150}{7}\right)\sqrt{3}\right)-\left(\dfrac{33}{8}+\left(-24\right)\sqrt{3}\right)-\left(\dfrac{231}{4}+\left(\dfrac{255}{2}\right)\sqrt{3}+\left(\dfrac{60}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{46}{3}\right)\sqrt{3}-\dfrac{153}{2}+\left(-\dfrac{25}{2}\right)\sqrt{3}+\left(-\dfrac{26}{9}\right)\sqrt{3}\right)\right)+\left(-\dfrac{159}{5}-\left(\left(-30\right)\sqrt{3}+\left(9\right)\sqrt{3}-\dfrac{159}{5}\right)+\dfrac{189}{8}-\left(\left(9\right)\sqrt{3}+\left(\dfrac{260}{9}\right)\sqrt{3}\right)\right)\\ &=&\left(\dfrac{231}{4}+\left(\dfrac{46}{3}\right)\sqrt{3}-\dfrac{159}{2}\right)-\left(\left(\dfrac{150}{7}\right)\sqrt{3}\right)-\left(\dfrac{33}{8}+\left(-24\right)\sqrt{3}\right)-\left(\dfrac{231}{4}+\left(\dfrac{255}{2}\right)\sqrt{3}+\left(\dfrac{60}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{46}{3}\right)\sqrt{3}-\dfrac{153}{2}+\left(-\dfrac{25}{2}\right)\sqrt{3}+\left(-\dfrac{26}{9}\right)\sqrt{3}\right)-\dfrac{159}{5}-\left(\left(-30\right)\sqrt{3}+\left(9\right)\sqrt{3}-\dfrac{159}{5}\right)+\dfrac{189}{8}-\left(\left(9\right)\sqrt{3}+\left(\dfrac{260}{9}\right)\sqrt{3}\right)\\ &=&\dfrac{33}{2}+\left(-135\right)\sqrt{3}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\dfrac{77}{4}\right)\sqrt{9}+\left(\dfrac{23}{3}\right)\sqrt{12}+\left(-\dfrac{53}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{9}+\left(-8\right)\sqrt{27}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{9}+\left(\dfrac{51}{2}\right)\sqrt{75}+\left(\dfrac{30}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{12}+\left(-\dfrac{51}{2}\right)\sqrt{9}+\left(-\dfrac{5}{2}\right)\sqrt{75}+\left(-\dfrac{13}{9}\right)\sqrt{12}\right)\right)-\left(\left(\left(-\dfrac{53}{5}\right)\sqrt{9}\right)-\left(\left(-6\right)\sqrt{75}+\left(3\right)\sqrt{27}+\left(-\dfrac{53}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{27}+\left(\dfrac{52}{9}\right)\sqrt{75}\right)\right)\\ &=&\left(\left(\dfrac{231}{4}+\left(\dfrac{46}{3}\right)\sqrt{3}-\dfrac{159}{2}\right)-\left(\left(\dfrac{150}{7}\right)\sqrt{3}\right)-\left(\dfrac{33}{8}+\left(-24\right)\sqrt{3}\right)-\left(\dfrac{231}{4}+\left(\dfrac{255}{2}\right)\sqrt{3}+\left(\dfrac{60}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{46}{3}\right)\sqrt{3}-\dfrac{153}{2}+\left(-\dfrac{25}{2}\right)\sqrt{3}+\left(-\dfrac{26}{9}\right)\sqrt{3}\right)\right)-\left(-\dfrac{159}{5}-\left(\left(-30\right)\sqrt{3}+\left(9\right)\sqrt{3}-\dfrac{159}{5}\right)+\dfrac{189}{8}-\left(\left(9\right)\sqrt{3}+\left(\dfrac{260}{9}\right)\sqrt{3}\right)\right)\\ &=&\left(-\dfrac{57}{8}+\left(-\dfrac{1063}{9}\right)\sqrt{3}\right)-\left(\dfrac{189}{8}+\left(-\dfrac{152}{9}\right)\sqrt{3}\right)\\ &=&-\dfrac{57}{8}+\left(-\dfrac{1063}{9}\right)\sqrt{3}+-\dfrac{189}{8}+\left(\dfrac{152}{9}\right)\sqrt{3}\\ &=&-\dfrac{123}{4}+\left(-\dfrac{911}{9}\right)\sqrt{3}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\dfrac{77}{4}\right)\sqrt{9}+\left(\dfrac{23}{3}\right)\sqrt{12}+\left(-\dfrac{53}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{9}+\left(-8\right)\sqrt{27}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{9}+\left(\dfrac{51}{2}\right)\sqrt{75}+\left(\dfrac{30}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{12}+\left(-\dfrac{51}{2}\right)\sqrt{9}+\left(-\dfrac{5}{2}\right)\sqrt{75}+\left(-\dfrac{13}{9}\right)\sqrt{12}\right)\right)\times\left(\left(\left(-\dfrac{53}{5}\right)\sqrt{9}\right)-\left(\left(-6\right)\sqrt{75}+\left(3\right)\sqrt{27}+\left(-\dfrac{53}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{27}+\left(\dfrac{52}{9}\right)\sqrt{75}\right)\right)\\ &=&\left(\left(\dfrac{231}{4}+\left(\dfrac{46}{3}\right)\sqrt{3}-\dfrac{159}{2}\right)-\left(\left(\dfrac{150}{7}\right)\sqrt{3}\right)-\left(\dfrac{33}{8}+\left(-24\right)\sqrt{3}\right)-\left(\dfrac{231}{4}+\left(\dfrac{255}{2}\right)\sqrt{3}+\left(\dfrac{60}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{46}{3}\right)\sqrt{3}-\dfrac{153}{2}+\left(-\dfrac{25}{2}\right)\sqrt{3}+\left(-\dfrac{26}{9}\right)\sqrt{3}\right)\right)\times\left(-\dfrac{159}{5}-\left(\left(-30\right)\sqrt{3}+\left(9\right)\sqrt{3}-\dfrac{159}{5}\right)+\dfrac{189}{8}-\left(\left(9\right)\sqrt{3}+\left(\dfrac{260}{9}\right)\sqrt{3}\right)\right)\\ &=&\left(-\dfrac{57}{8}+\left(-\dfrac{1063}{9}\right)\sqrt{3}\right)\left(\dfrac{189}{8}+\left(-\dfrac{152}{9}\right)\sqrt{3}\right)\\ &=&-\dfrac{10773}{64}+\left(-\dfrac{64081}{24}\right)\sqrt{3}+\left(\dfrac{161576}{81}\right)\sqrt{9}\\ &=&\dfrac{10049993}{1728}+\left(-\dfrac{64081}{24}\right)\sqrt{3}\\ \end{eqnarray*}