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{23}{2}-\left(\left(-2\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{9}\right)\sqrt{125}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{125}\right)-\left(\left(-9\right)\sqrt{45}\right)\right)-\left(\left(9\right)\sqrt{25}+\left(2\right)\sqrt{25}+\left(-1\right)\sqrt{20}+\left(\dfrac{26}{3}\right)\sqrt{25}+\left(\dfrac{17}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{51}{4}\right)\sqrt{25}+\left(\dfrac{72}{7}\right)\sqrt{25}+\left(-\dfrac{67}{9}\right)\sqrt{125}+\left(-2\right)\sqrt{25}\right)$ et $ Y=\left(\dfrac{59}{4}-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)\right)-\left(\left(-\dfrac{45}{4}\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{14}{5}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{58}{5}\right)\sqrt{125}\right)-\left(\left(3\right)\sqrt{125}\right)\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(\dfrac{23}{2}-\left(\left(-2\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{9}\right)\sqrt{125}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{125}\right)-\left(\left(-9\right)\sqrt{45}\right)\right)-\left(\left(9\right)\sqrt{25}+\left(2\right)\sqrt{25}+\left(-1\right)\sqrt{20}+\left(\dfrac{26}{3}\right)\sqrt{25}+\left(\dfrac{17}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{51}{4}\right)\sqrt{25}+\left(\dfrac{72}{7}\right)\sqrt{25}+\left(-\dfrac{67}{9}\right)\sqrt{125}+\left(-2\right)\sqrt{25}\right)\right)+\left(\left(\dfrac{59}{4}-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)\right)-\left(\left(-\dfrac{45}{4}\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{14}{5}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{58}{5}\right)\sqrt{125}\right)-\left(\left(3\right)\sqrt{125}\right)\right)\right)\\ &=&\left(\left(\dfrac{23}{2}+10-\left(\left(\dfrac{55}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{245}{3}\right)\sqrt{5}\right)-\left(\left(-27\right)\sqrt{5}\right)\right)-\left(45+10+\left(-2\right)\sqrt{5}+\dfrac{130}{3}+17\right)-\left(\dfrac{255}{4}+\dfrac{360}{7}+\left(-\dfrac{335}{9}\right)\sqrt{5}-10\right)\right)+\left(\left(\dfrac{59}{4}-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{135}{4}\right)\sqrt{5}\right)-\left(14-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(58\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)\right)\right)\\ &=&\left(\dfrac{23}{2}+10-\left(\left(\dfrac{55}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{245}{3}\right)\sqrt{5}\right)-\left(\left(-27\right)\sqrt{5}\right)\right)-\left(45+10+\left(-2\right)\sqrt{5}+\dfrac{130}{3}+17\right)-\left(\dfrac{255}{4}+\dfrac{360}{7}+\left(-\dfrac{335}{9}\right)\sqrt{5}-10\right)+\left(\dfrac{59}{4}-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{135}{4}\right)\sqrt{5}\right)-\left(14-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(58\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)\right)\\ &=&-\dfrac{8327}{42}+\left(\dfrac{8569}{36}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\dfrac{23}{2}-\left(\left(-2\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{9}\right)\sqrt{125}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{125}\right)-\left(\left(-9\right)\sqrt{45}\right)\right)-\left(\left(9\right)\sqrt{25}+\left(2\right)\sqrt{25}+\left(-1\right)\sqrt{20}+\left(\dfrac{26}{3}\right)\sqrt{25}+\left(\dfrac{17}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{51}{4}\right)\sqrt{25}+\left(\dfrac{72}{7}\right)\sqrt{25}+\left(-\dfrac{67}{9}\right)\sqrt{125}+\left(-2\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{59}{4}-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)\right)-\left(\left(-\dfrac{45}{4}\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{14}{5}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{58}{5}\right)\sqrt{125}\right)-\left(\left(3\right)\sqrt{125}\right)\right)\right)\\ &=&\left(\left(\dfrac{23}{2}+10-\left(\left(\dfrac{55}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{245}{3}\right)\sqrt{5}\right)-\left(\left(-27\right)\sqrt{5}\right)\right)-\left(45+10+\left(-2\right)\sqrt{5}+\dfrac{130}{3}+17\right)-\left(\dfrac{255}{4}+\dfrac{360}{7}+\left(-\dfrac{335}{9}\right)\sqrt{5}-10\right)\right)-\left(\left(\dfrac{59}{4}-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{135}{4}\right)\sqrt{5}\right)-\left(14-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(58\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)\right)\right)\\ &=&\left(-\dfrac{16717}{84}+\left(\dfrac{1276}{9}\right)\sqrt{5}\right)-\left(\dfrac{3}{4}+\left(\dfrac{385}{4}\right)\sqrt{5}\right)\\ &=&-\dfrac{16717}{84}+\left(\dfrac{1276}{9}\right)\sqrt{5}+-\dfrac{3}{4}+\left(-\dfrac{385}{4}\right)\sqrt{5}\\ &=&-\dfrac{4195}{21}+\left(\dfrac{1639}{36}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\dfrac{23}{2}-\left(\left(-2\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{9}\right)\sqrt{125}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{125}\right)-\left(\left(-9\right)\sqrt{45}\right)\right)-\left(\left(9\right)\sqrt{25}+\left(2\right)\sqrt{25}+\left(-1\right)\sqrt{20}+\left(\dfrac{26}{3}\right)\sqrt{25}+\left(\dfrac{17}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{51}{4}\right)\sqrt{25}+\left(\dfrac{72}{7}\right)\sqrt{25}+\left(-\dfrac{67}{9}\right)\sqrt{125}+\left(-2\right)\sqrt{25}\right)\right)\times\left(\left(\dfrac{59}{4}-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)\right)-\left(\left(-\dfrac{45}{4}\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{14}{5}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{58}{5}\right)\sqrt{125}\right)-\left(\left(3\right)\sqrt{125}\right)\right)\right)\\ &=&\left(\left(\dfrac{23}{2}+10-\left(\left(\dfrac{55}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{245}{3}\right)\sqrt{5}\right)-\left(\left(-27\right)\sqrt{5}\right)\right)-\left(45+10+\left(-2\right)\sqrt{5}+\dfrac{130}{3}+17\right)-\left(\dfrac{255}{4}+\dfrac{360}{7}+\left(-\dfrac{335}{9}\right)\sqrt{5}-10\right)\right)\times\left(\left(\dfrac{59}{4}-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{135}{4}\right)\sqrt{5}\right)-\left(14-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(58\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)\right)\right)\\ &=&\left(-\dfrac{16717}{84}+\left(\dfrac{1276}{9}\right)\sqrt{5}\right)\left(\dfrac{3}{4}+\left(\dfrac{385}{4}\right)\sqrt{5}\right)\\ &=&-\dfrac{16717}{112}+\left(-\dfrac{304777}{16}\right)\sqrt{5}+\left(\dfrac{122815}{9}\right)\sqrt{25}\\ &=&\dfrac{68625947}{1008}+\left(-\dfrac{304777}{16}\right)\sqrt{5}\\ \end{eqnarray*}