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{17}{4}+\left(-\dfrac{26}{3}\right)\sqrt{63}+\left(-\dfrac{67}{9}\right)\sqrt{63}+\left(\dfrac{58}{3}\right)\sqrt{28}+\dfrac{33}{2}+\left(-\dfrac{61}{7}\right)\sqrt{28}$ et $ Y=\left(\left(\dfrac{56}{3}\right)\sqrt{28}\right)-\left(\left(-7\right)\sqrt{49}\right)-6+9-\left(\left(5\right)\sqrt{175}\right)+\left(\dfrac{74}{9}\right)\sqrt{63}+5+\left(-\dfrac{7}{3}\right)\sqrt{49}-\dfrac{19}{4}+\left(-4\right)\sqrt{63}+\left(-\dfrac{51}{2}\right)\sqrt{28}+\left(\left(0\right)\sqrt{63}\right)-\left(\left(\dfrac{29}{8}\right)\sqrt{175}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{175}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{68}{3}\right)\sqrt{28}\right)+\left(\dfrac{53}{8}\right)\sqrt{28}$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(\dfrac{17}{4}+\left(-\dfrac{26}{3}\right)\sqrt{63}+\left(-\dfrac{67}{9}\right)\sqrt{63}+\left(\dfrac{58}{3}\right)\sqrt{28}+\dfrac{33}{2}+\left(-\dfrac{61}{7}\right)\sqrt{28}\right)+\left(\left(\left(\dfrac{56}{3}\right)\sqrt{28}\right)-\left(\left(-7\right)\sqrt{49}\right)-6+9-\left(\left(5\right)\sqrt{175}\right)+\left(\dfrac{74}{9}\right)\sqrt{63}+5+\left(-\dfrac{7}{3}\right)\sqrt{49}-\dfrac{19}{4}+\left(-4\right)\sqrt{63}+\left(-\dfrac{51}{2}\right)\sqrt{28}+\left(\left(0\right)\sqrt{63}\right)-\left(\left(\dfrac{29}{8}\right)\sqrt{175}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{175}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{68}{3}\right)\sqrt{28}\right)+\left(\dfrac{53}{8}\right)\sqrt{28}\right)\\ &=&\left(\dfrac{17}{4}+\left(-26\right)\sqrt{7}+\left(-\dfrac{67}{3}\right)\sqrt{7}+\left(\dfrac{116}{3}\right)\sqrt{7}+\dfrac{33}{2}+\left(-\dfrac{122}{7}\right)\sqrt{7}\right)+\left(\left(\left(\dfrac{112}{3}\right)\sqrt{7}\right)+49-6+9-\left(\left(25\right)\sqrt{7}\right)+\left(\dfrac{74}{3}\right)\sqrt{7}+5-\dfrac{49}{3}-\dfrac{19}{4}+\left(-12\right)\sqrt{7}+\left(-51\right)\sqrt{7}+\left(\left(0\right)\sqrt{7}\right)-\left(\left(\dfrac{145}{8}\right)\sqrt{7}\right)-\left(\left(-\dfrac{195}{2}\right)\sqrt{7}\right)-\dfrac{105}{4}-\left(\left(-\dfrac{136}{3}\right)\sqrt{7}\right)+\left(\dfrac{53}{4}\right)\sqrt{7}\right)\\ &=&\dfrac{17}{4}+\left(-26\right)\sqrt{7}+\left(-\dfrac{67}{3}\right)\sqrt{7}+\left(\dfrac{116}{3}\right)\sqrt{7}+\dfrac{33}{2}+\left(-\dfrac{122}{7}\right)\sqrt{7}+\left(\left(\dfrac{112}{3}\right)\sqrt{7}\right)+49-6+9-\left(\left(25\right)\sqrt{7}\right)+\left(\dfrac{74}{3}\right)\sqrt{7}+5-\dfrac{49}{3}-\dfrac{19}{4}+\left(-12\right)\sqrt{7}+\left(-51\right)\sqrt{7}+\left(\left(0\right)\sqrt{7}\right)-\left(\left(\dfrac{145}{8}\right)\sqrt{7}\right)-\left(\left(-\dfrac{195}{2}\right)\sqrt{7}\right)-\dfrac{105}{4}-\left(\left(-\dfrac{136}{3}\right)\sqrt{7}\right)+\left(\dfrac{53}{4}\right)\sqrt{7}\\ &=&\dfrac{365}{12}+\left(\dfrac{14257}{168}\right)\sqrt{7}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\dfrac{17}{4}+\left(-\dfrac{26}{3}\right)\sqrt{63}+\left(-\dfrac{67}{9}\right)\sqrt{63}+\left(\dfrac{58}{3}\right)\sqrt{28}+\dfrac{33}{2}+\left(-\dfrac{61}{7}\right)\sqrt{28}\right)-\left(\left(\left(\dfrac{56}{3}\right)\sqrt{28}\right)-\left(\left(-7\right)\sqrt{49}\right)-6+9-\left(\left(5\right)\sqrt{175}\right)+\left(\dfrac{74}{9}\right)\sqrt{63}+5+\left(-\dfrac{7}{3}\right)\sqrt{49}-\dfrac{19}{4}+\left(-4\right)\sqrt{63}+\left(-\dfrac{51}{2}\right)\sqrt{28}+\left(\left(0\right)\sqrt{63}\right)-\left(\left(\dfrac{29}{8}\right)\sqrt{175}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{175}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{68}{3}\right)\sqrt{28}\right)+\left(\dfrac{53}{8}\right)\sqrt{28}\right)\\ &=&\left(\dfrac{17}{4}+\left(-26\right)\sqrt{7}+\left(-\dfrac{67}{3}\right)\sqrt{7}+\left(\dfrac{116}{3}\right)\sqrt{7}+\dfrac{33}{2}+\left(-\dfrac{122}{7}\right)\sqrt{7}\right)-\left(\left(\left(\dfrac{112}{3}\right)\sqrt{7}\right)+49-6+9-\left(\left(25\right)\sqrt{7}\right)+\left(\dfrac{74}{3}\right)\sqrt{7}+5-\dfrac{49}{3}-\dfrac{19}{4}+\left(-12\right)\sqrt{7}+\left(-51\right)\sqrt{7}+\left(\left(0\right)\sqrt{7}\right)-\left(\left(\dfrac{145}{8}\right)\sqrt{7}\right)-\left(\left(-\dfrac{195}{2}\right)\sqrt{7}\right)-\dfrac{105}{4}-\left(\left(-\dfrac{136}{3}\right)\sqrt{7}\right)+\left(\dfrac{53}{4}\right)\sqrt{7}\right)\\ &=&\left(\dfrac{83}{4}+\left(-\dfrac{569}{21}\right)\sqrt{7}\right)-\left(\left(\dfrac{2687}{24}\right)\sqrt{7}+\dfrac{29}{3}\right)\\ &=&\dfrac{83}{4}+\left(-\dfrac{569}{21}\right)\sqrt{7}+\left(-\dfrac{2687}{24}\right)\sqrt{7}-\dfrac{29}{3}\\ &=&\dfrac{133}{12}+\left(-\dfrac{7787}{56}\right)\sqrt{7}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\dfrac{17}{4}+\left(-\dfrac{26}{3}\right)\sqrt{63}+\left(-\dfrac{67}{9}\right)\sqrt{63}+\left(\dfrac{58}{3}\right)\sqrt{28}+\dfrac{33}{2}+\left(-\dfrac{61}{7}\right)\sqrt{28}\right)\times\left(\left(\left(\dfrac{56}{3}\right)\sqrt{28}\right)-\left(\left(-7\right)\sqrt{49}\right)-6+9-\left(\left(5\right)\sqrt{175}\right)+\left(\dfrac{74}{9}\right)\sqrt{63}+5+\left(-\dfrac{7}{3}\right)\sqrt{49}-\dfrac{19}{4}+\left(-4\right)\sqrt{63}+\left(-\dfrac{51}{2}\right)\sqrt{28}+\left(\left(0\right)\sqrt{63}\right)-\left(\left(\dfrac{29}{8}\right)\sqrt{175}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{175}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{68}{3}\right)\sqrt{28}\right)+\left(\dfrac{53}{8}\right)\sqrt{28}\right)\\ &=&\left(\dfrac{17}{4}+\left(-26\right)\sqrt{7}+\left(-\dfrac{67}{3}\right)\sqrt{7}+\left(\dfrac{116}{3}\right)\sqrt{7}+\dfrac{33}{2}+\left(-\dfrac{122}{7}\right)\sqrt{7}\right)\times\left(\left(\left(\dfrac{112}{3}\right)\sqrt{7}\right)+49-6+9-\left(\left(25\right)\sqrt{7}\right)+\left(\dfrac{74}{3}\right)\sqrt{7}+5-\dfrac{49}{3}-\dfrac{19}{4}+\left(-12\right)\sqrt{7}+\left(-51\right)\sqrt{7}+\left(\left(0\right)\sqrt{7}\right)-\left(\left(\dfrac{145}{8}\right)\sqrt{7}\right)-\left(\left(-\dfrac{195}{2}\right)\sqrt{7}\right)-\dfrac{105}{4}-\left(\left(-\dfrac{136}{3}\right)\sqrt{7}\right)+\left(\dfrac{53}{4}\right)\sqrt{7}\right)\\ &=&\left(\dfrac{83}{4}+\left(-\dfrac{569}{21}\right)\sqrt{7}\right)\left(\left(\dfrac{2687}{24}\right)\sqrt{7}+\dfrac{29}{3}\right)\\ &=&\left(\dfrac{4155409}{2016}\right)\sqrt{7}+\dfrac{2407}{12}+\left(-\dfrac{1528903}{504}\right)\sqrt{49}\\ &=&\left(\dfrac{4155409}{2016}\right)\sqrt{7}-\dfrac{1514461}{72}\\ \end{eqnarray*}