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(\dfrac{63}{8}\right)\sqrt{27}+\left(-\dfrac{15}{2}\right)\sqrt{75}+\left(-7\right)\sqrt{12}+\left(\left(-\dfrac{53}{8}\right)\sqrt{12}\right)-\left(\left(\dfrac{41}{6}\right)\sqrt{12}\right)+\left(\dfrac{65}{4}\right)\sqrt{75}+\left(-9\right)\sqrt{12}+4+\left(\dfrac{21}{2}\right)\sqrt{9}+\dfrac{17}{8}+\left(\dfrac{11}{4}\right)\sqrt{9}+\left(\left(-\dfrac{80}{9}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(-\dfrac{65}{6}\right)\sqrt{27}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)$ et $ Y=\left(\left(-4\right)\sqrt{27}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{75}\right)-\left(\left(-2\right)\sqrt{27}\right)+\left(\dfrac{17}{3}\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(\dfrac{63}{8}\right)\sqrt{27}+\left(-\dfrac{15}{2}\right)\sqrt{75}+\left(-7\right)\sqrt{12}+\left(\left(-\dfrac{53}{8}\right)\sqrt{12}\right)-\left(\left(\dfrac{41}{6}\right)\sqrt{12}\right)+\left(\dfrac{65}{4}\right)\sqrt{75}+\left(-9\right)\sqrt{12}+4+\left(\dfrac{21}{2}\right)\sqrt{9}+\dfrac{17}{8}+\left(\dfrac{11}{4}\right)\sqrt{9}+\left(\left(-\dfrac{80}{9}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(-\dfrac{65}{6}\right)\sqrt{27}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)\right)+\left(\left(\left(-4\right)\sqrt{27}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{75}\right)-\left(\left(-2\right)\sqrt{27}\right)+\left(\dfrac{17}{3}\right)\sqrt{12}\right)\\ &=&\left(\left(\dfrac{189}{8}\right)\sqrt{3}+\left(-\dfrac{75}{2}\right)\sqrt{3}+\left(-14\right)\sqrt{3}+\left(\left(-\dfrac{53}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{41}{3}\right)\sqrt{3}\right)+\left(\dfrac{325}{4}\right)\sqrt{3}+\left(-18\right)\sqrt{3}+4+\dfrac{63}{2}+\dfrac{17}{8}+\dfrac{33}{4}+\left(\left(-\dfrac{160}{9}\right)\sqrt{3}\right)+24-\left(\left(-\dfrac{65}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)\right)+\left(\left(\left(-12\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)+\left(\dfrac{34}{3}\right)\sqrt{3}\right)\\ &=&\left(\dfrac{189}{8}\right)\sqrt{3}+\left(-\dfrac{75}{2}\right)\sqrt{3}+\left(-14\right)\sqrt{3}+\left(\left(-\dfrac{53}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{41}{3}\right)\sqrt{3}\right)+\left(\dfrac{325}{4}\right)\sqrt{3}+\left(-18\right)\sqrt{3}+4+\dfrac{63}{2}+\dfrac{17}{8}+\dfrac{33}{4}+\left(\left(-\dfrac{160}{9}\right)\sqrt{3}\right)+24-\left(\left(-\dfrac{65}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)+\left(\left(-12\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)+\left(\dfrac{34}{3}\right)\sqrt{3}\\ &=&\left(\dfrac{5539}{72}\right)\sqrt{3}+\dfrac{559}{8}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\dfrac{63}{8}\right)\sqrt{27}+\left(-\dfrac{15}{2}\right)\sqrt{75}+\left(-7\right)\sqrt{12}+\left(\left(-\dfrac{53}{8}\right)\sqrt{12}\right)-\left(\left(\dfrac{41}{6}\right)\sqrt{12}\right)+\left(\dfrac{65}{4}\right)\sqrt{75}+\left(-9\right)\sqrt{12}+4+\left(\dfrac{21}{2}\right)\sqrt{9}+\dfrac{17}{8}+\left(\dfrac{11}{4}\right)\sqrt{9}+\left(\left(-\dfrac{80}{9}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(-\dfrac{65}{6}\right)\sqrt{27}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)\right)-\left(\left(\left(-4\right)\sqrt{27}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{75}\right)-\left(\left(-2\right)\sqrt{27}\right)+\left(\dfrac{17}{3}\right)\sqrt{12}\right)\\ &=&\left(\left(\dfrac{189}{8}\right)\sqrt{3}+\left(-\dfrac{75}{2}\right)\sqrt{3}+\left(-14\right)\sqrt{3}+\left(\left(-\dfrac{53}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{41}{3}\right)\sqrt{3}\right)+\left(\dfrac{325}{4}\right)\sqrt{3}+\left(-18\right)\sqrt{3}+4+\dfrac{63}{2}+\dfrac{17}{8}+\dfrac{33}{4}+\left(\left(-\dfrac{160}{9}\right)\sqrt{3}\right)+24-\left(\left(-\dfrac{65}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)\right)-\left(\left(\left(-12\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)+\left(\dfrac{34}{3}\right)\sqrt{3}\right)\\ &=&\left(\left(\dfrac{4345}{72}\right)\sqrt{3}+\dfrac{559}{8}\right)-\left(\left(\dfrac{199}{12}\right)\sqrt{3}\right)\\ &=&\left(\dfrac{4345}{72}\right)\sqrt{3}+\dfrac{559}{8}+\left(-\dfrac{199}{12}\right)\sqrt{3}\\ &=&\left(\dfrac{3151}{72}\right)\sqrt{3}+\dfrac{559}{8}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\dfrac{63}{8}\right)\sqrt{27}+\left(-\dfrac{15}{2}\right)\sqrt{75}+\left(-7\right)\sqrt{12}+\left(\left(-\dfrac{53}{8}\right)\sqrt{12}\right)-\left(\left(\dfrac{41}{6}\right)\sqrt{12}\right)+\left(\dfrac{65}{4}\right)\sqrt{75}+\left(-9\right)\sqrt{12}+4+\left(\dfrac{21}{2}\right)\sqrt{9}+\dfrac{17}{8}+\left(\dfrac{11}{4}\right)\sqrt{9}+\left(\left(-\dfrac{80}{9}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(-\dfrac{65}{6}\right)\sqrt{27}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)\right)\times\left(\left(\left(-4\right)\sqrt{27}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{75}\right)-\left(\left(-2\right)\sqrt{27}\right)+\left(\dfrac{17}{3}\right)\sqrt{12}\right)\\ &=&\left(\left(\dfrac{189}{8}\right)\sqrt{3}+\left(-\dfrac{75}{2}\right)\sqrt{3}+\left(-14\right)\sqrt{3}+\left(\left(-\dfrac{53}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{41}{3}\right)\sqrt{3}\right)+\left(\dfrac{325}{4}\right)\sqrt{3}+\left(-18\right)\sqrt{3}+4+\dfrac{63}{2}+\dfrac{17}{8}+\dfrac{33}{4}+\left(\left(-\dfrac{160}{9}\right)\sqrt{3}\right)+24-\left(\left(-\dfrac{65}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)\right)\times\left(\left(\left(-12\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)+\left(\dfrac{34}{3}\right)\sqrt{3}\right)\\ &=&\left(\left(\dfrac{4345}{72}\right)\sqrt{3}+\dfrac{559}{8}\right)\left(\left(\dfrac{199}{12}\right)\sqrt{3}\right)\\ &=&\left(\dfrac{864655}{864}\right)\sqrt{9}+\left(\dfrac{111241}{96}\right)\sqrt{3}\\ &=&\dfrac{864655}{288}+\left(\dfrac{111241}{96}\right)\sqrt{3}\\ \end{eqnarray*}