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{29}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{75}+\left(\dfrac{31}{5}\right)\sqrt{27}+\left(-\dfrac{48}{7}\right)\sqrt{75}\right)-\left(\dfrac{78}{5}+\left(\dfrac{11}{2}\right)\sqrt{9}+\left(-\dfrac{7}{4}\right)\sqrt{75}+\left(-\dfrac{76}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{75}\right)$ et $ Y=\left(-9\right)\sqrt{27}+\left(\dfrac{27}{2}\right)\sqrt{75}+\left(-\dfrac{7}{5}\right)\sqrt{12}+\dfrac{58}{5}-\dfrac{34}{9}+\left(0\right)\sqrt{27}+\left(-\dfrac{13}{2}\right)\sqrt{9}+\left(\dfrac{40}{7}\right)\sqrt{27}+\left(\left(-\dfrac{13}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{68}{9}\right)\sqrt{9}\right)+1-\left(\left(-\dfrac{13}{2}\right)\sqrt{9}\right)-\dfrac{67}{3}+\left(-9\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{29}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{75}+\left(\dfrac{31}{5}\right)\sqrt{27}+\left(-\dfrac{48}{7}\right)\sqrt{75}\right)-\left(\dfrac{78}{5}+\left(\dfrac{11}{2}\right)\sqrt{9}+\left(-\dfrac{7}{4}\right)\sqrt{75}+\left(-\dfrac{76}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{75}\right)\right)+\left(\left(-9\right)\sqrt{27}+\left(\dfrac{27}{2}\right)\sqrt{75}+\left(-\dfrac{7}{5}\right)\sqrt{12}+\dfrac{58}{5}-\dfrac{34}{9}+\left(0\right)\sqrt{27}+\left(-\dfrac{13}{2}\right)\sqrt{9}+\left(\dfrac{40}{7}\right)\sqrt{27}+\left(\left(-\dfrac{13}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{68}{9}\right)\sqrt{9}\right)+1-\left(\left(-\dfrac{13}{2}\right)\sqrt{9}\right)-\dfrac{67}{3}+\left(-9\right)\sqrt{27}\right)\\ &=&\left(\left(\left(-\dfrac{145}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{370}{7}\right)\sqrt{3}+\left(\dfrac{93}{5}\right)\sqrt{3}+\left(-\dfrac{240}{7}\right)\sqrt{3}\right)-\left(\dfrac{78}{5}+\dfrac{33}{2}+\left(-\dfrac{35}{4}\right)\sqrt{3}+\left(-\dfrac{152}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{315}{4}\right)\sqrt{3}\right)\right)+\left(\left(-27\right)\sqrt{3}+\left(\dfrac{135}{2}\right)\sqrt{3}+\left(-\dfrac{14}{5}\right)\sqrt{3}+\dfrac{58}{5}-\dfrac{34}{9}+\left(0\right)\sqrt{3}-\dfrac{39}{2}+\left(\dfrac{120}{7}\right)\sqrt{3}-\dfrac{39}{2}-\dfrac{68}{3}+1+\dfrac{39}{2}-\dfrac{67}{3}+\left(-27\right)\sqrt{3}\right)\\ &=&\left(\left(-\dfrac{145}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{370}{7}\right)\sqrt{3}+\left(\dfrac{93}{5}\right)\sqrt{3}+\left(-\dfrac{240}{7}\right)\sqrt{3}\right)-\left(\dfrac{78}{5}+\dfrac{33}{2}+\left(-\dfrac{35}{4}\right)\sqrt{3}+\left(-\dfrac{152}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{315}{4}\right)\sqrt{3}\right)+\left(-27\right)\sqrt{3}+\left(\dfrac{135}{2}\right)\sqrt{3}+\left(-\dfrac{14}{5}\right)\sqrt{3}+\dfrac{58}{5}-\dfrac{34}{9}+\left(0\right)\sqrt{3}-\dfrac{39}{2}+\left(\dfrac{120}{7}\right)\sqrt{3}-\dfrac{39}{2}-\dfrac{68}{3}+1+\dfrac{39}{2}-\dfrac{67}{3}+\left(-27\right)\sqrt{3}\\ &=&\left(\dfrac{1731}{28}\right)\sqrt{3}-\dfrac{790}{9}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(-\dfrac{29}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{75}+\left(\dfrac{31}{5}\right)\sqrt{27}+\left(-\dfrac{48}{7}\right)\sqrt{75}\right)-\left(\dfrac{78}{5}+\left(\dfrac{11}{2}\right)\sqrt{9}+\left(-\dfrac{7}{4}\right)\sqrt{75}+\left(-\dfrac{76}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{75}\right)\right)-\left(\left(-9\right)\sqrt{27}+\left(\dfrac{27}{2}\right)\sqrt{75}+\left(-\dfrac{7}{5}\right)\sqrt{12}+\dfrac{58}{5}-\dfrac{34}{9}+\left(0\right)\sqrt{27}+\left(-\dfrac{13}{2}\right)\sqrt{9}+\left(\dfrac{40}{7}\right)\sqrt{27}+\left(\left(-\dfrac{13}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{68}{9}\right)\sqrt{9}\right)+1-\left(\left(-\dfrac{13}{2}\right)\sqrt{9}\right)-\dfrac{67}{3}+\left(-9\right)\sqrt{27}\right)\\ &=&\left(\left(\left(-\dfrac{145}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{370}{7}\right)\sqrt{3}+\left(\dfrac{93}{5}\right)\sqrt{3}+\left(-\dfrac{240}{7}\right)\sqrt{3}\right)-\left(\dfrac{78}{5}+\dfrac{33}{2}+\left(-\dfrac{35}{4}\right)\sqrt{3}+\left(-\dfrac{152}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{315}{4}\right)\sqrt{3}\right)\right)-\left(\left(-27\right)\sqrt{3}+\left(\dfrac{135}{2}\right)\sqrt{3}+\left(-\dfrac{14}{5}\right)\sqrt{3}+\dfrac{58}{5}-\dfrac{34}{9}+\left(0\right)\sqrt{3}-\dfrac{39}{2}+\left(\dfrac{120}{7}\right)\sqrt{3}-\dfrac{39}{2}-\dfrac{68}{3}+1+\dfrac{39}{2}-\dfrac{67}{3}+\left(-27\right)\sqrt{3}\right)\\ &=&\left(\left(\dfrac{4757}{140}\right)\sqrt{3}-\dfrac{321}{10}\right)-\left(\left(\dfrac{1949}{70}\right)\sqrt{3}-\dfrac{5011}{90}\right)\\ &=&\left(\dfrac{4757}{140}\right)\sqrt{3}-\dfrac{321}{10}+\left(-\dfrac{1949}{70}\right)\sqrt{3}+\dfrac{5011}{90}\\ &=&\left(\dfrac{859}{140}\right)\sqrt{3}+\dfrac{1061}{45}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(-\dfrac{29}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{75}+\left(\dfrac{31}{5}\right)\sqrt{27}+\left(-\dfrac{48}{7}\right)\sqrt{75}\right)-\left(\dfrac{78}{5}+\left(\dfrac{11}{2}\right)\sqrt{9}+\left(-\dfrac{7}{4}\right)\sqrt{75}+\left(-\dfrac{76}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{75}\right)\right)\times\left(\left(-9\right)\sqrt{27}+\left(\dfrac{27}{2}\right)\sqrt{75}+\left(-\dfrac{7}{5}\right)\sqrt{12}+\dfrac{58}{5}-\dfrac{34}{9}+\left(0\right)\sqrt{27}+\left(-\dfrac{13}{2}\right)\sqrt{9}+\left(\dfrac{40}{7}\right)\sqrt{27}+\left(\left(-\dfrac{13}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{68}{9}\right)\sqrt{9}\right)+1-\left(\left(-\dfrac{13}{2}\right)\sqrt{9}\right)-\dfrac{67}{3}+\left(-9\right)\sqrt{27}\right)\\ &=&\left(\left(\left(-\dfrac{145}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{370}{7}\right)\sqrt{3}+\left(\dfrac{93}{5}\right)\sqrt{3}+\left(-\dfrac{240}{7}\right)\sqrt{3}\right)-\left(\dfrac{78}{5}+\dfrac{33}{2}+\left(-\dfrac{35}{4}\right)\sqrt{3}+\left(-\dfrac{152}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{315}{4}\right)\sqrt{3}\right)\right)\times\left(\left(-27\right)\sqrt{3}+\left(\dfrac{135}{2}\right)\sqrt{3}+\left(-\dfrac{14}{5}\right)\sqrt{3}+\dfrac{58}{5}-\dfrac{34}{9}+\left(0\right)\sqrt{3}-\dfrac{39}{2}+\left(\dfrac{120}{7}\right)\sqrt{3}-\dfrac{39}{2}-\dfrac{68}{3}+1+\dfrac{39}{2}-\dfrac{67}{3}+\left(-27\right)\sqrt{3}\right)\\ &=&\left(\left(\dfrac{4757}{140}\right)\sqrt{3}-\dfrac{321}{10}\right)\left(\left(\dfrac{1949}{70}\right)\sqrt{3}-\dfrac{5011}{90}\right)\\ &=&\left(\dfrac{9271393}{9800}\right)\sqrt{9}+\left(-\dfrac{24569054300}{8820000}\right)\sqrt{3}+\dfrac{536177}{300}\\ &=&\dfrac{13598788300}{2940000}+\left(-\dfrac{24569054300}{8820000}\right)\sqrt{3}\\ \end{eqnarray*}