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(5-\left(\left(2\right)\sqrt{4}\right)-\left(\left(4\right)\sqrt{18}\right)\right)-\left(\left(\left(-\dfrac{67}{5}\right)\sqrt{8}\right)-\left(\left(-\dfrac{33}{4}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{4}+\left(\dfrac{1}{5}\right)\sqrt{4}+\left(-\dfrac{59}{3}\right)\sqrt{50}\right)$ et $ Y=\left(\dfrac{65}{2}\right)\sqrt{50}-\dfrac{64}{3}-\left(\left(-2\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{4}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{4}\right)+\left(\dfrac{55}{3}\right)\sqrt{18}+\left(\dfrac{22}{3}\right)\sqrt{8}+\left(\left(\dfrac{65}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{8}\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(5-\left(\left(2\right)\sqrt{4}\right)-\left(\left(4\right)\sqrt{18}\right)\right)-\left(\left(\left(-\dfrac{67}{5}\right)\sqrt{8}\right)-\left(\left(-\dfrac{33}{4}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{4}+\left(\dfrac{1}{5}\right)\sqrt{4}+\left(-\dfrac{59}{3}\right)\sqrt{50}\right)\right)+\left(\left(\dfrac{65}{2}\right)\sqrt{50}-\dfrac{64}{3}-\left(\left(-2\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{4}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{4}\right)+\left(\dfrac{55}{3}\right)\sqrt{18}+\left(\dfrac{22}{3}\right)\sqrt{8}+\left(\left(\dfrac{65}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{8}\right)\right)\\ &=&\left(\left(5-4-\left(\left(12\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{134}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{165}{4}\right)\sqrt{2}\right)\right)-\dfrac{16}{3}-\left(-\dfrac{52}{7}+\dfrac{2}{5}+\left(-\dfrac{295}{3}\right)\sqrt{2}\right)\right)+\left(\left(\dfrac{325}{2}\right)\sqrt{2}-\dfrac{64}{3}-\left(\left(-4\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{4}\right)\sqrt{2}\right)+\dfrac{82}{7}+\left(55\right)\sqrt{2}+\left(\dfrac{44}{3}\right)\sqrt{2}+\left(\left(\dfrac{325}{2}\right)\sqrt{2}\right)+\dfrac{82}{7}-\left(\left(-\dfrac{86}{7}\right)\sqrt{2}\right)\right)\\ &=&\left(5-4-\left(\left(12\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{134}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{165}{4}\right)\sqrt{2}\right)\right)-\dfrac{16}{3}-\left(-\dfrac{52}{7}+\dfrac{2}{5}+\left(-\dfrac{295}{3}\right)\sqrt{2}\right)+\left(\dfrac{325}{2}\right)\sqrt{2}-\dfrac{64}{3}-\left(\left(-4\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{4}\right)\sqrt{2}\right)+\dfrac{82}{7}+\left(55\right)\sqrt{2}+\left(\dfrac{44}{3}\right)\sqrt{2}+\left(\left(\dfrac{325}{2}\right)\sqrt{2}\right)+\dfrac{82}{7}-\left(\left(-\dfrac{86}{7}\right)\sqrt{2}\right)\\ &=&\dfrac{503}{105}+\left(\dfrac{32661}{70}\right)\sqrt{2}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(5-\left(\left(2\right)\sqrt{4}\right)-\left(\left(4\right)\sqrt{18}\right)\right)-\left(\left(\left(-\dfrac{67}{5}\right)\sqrt{8}\right)-\left(\left(-\dfrac{33}{4}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{4}+\left(\dfrac{1}{5}\right)\sqrt{4}+\left(-\dfrac{59}{3}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{65}{2}\right)\sqrt{50}-\dfrac{64}{3}-\left(\left(-2\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{4}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{4}\right)+\left(\dfrac{55}{3}\right)\sqrt{18}+\left(\dfrac{22}{3}\right)\sqrt{8}+\left(\left(\dfrac{65}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{8}\right)\right)\\ &=&\left(\left(5-4-\left(\left(12\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{134}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{165}{4}\right)\sqrt{2}\right)\right)-\dfrac{16}{3}-\left(-\dfrac{52}{7}+\dfrac{2}{5}+\left(-\dfrac{295}{3}\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{325}{2}\right)\sqrt{2}-\dfrac{64}{3}-\left(\left(-4\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{4}\right)\sqrt{2}\right)+\dfrac{82}{7}+\left(55\right)\sqrt{2}+\left(\dfrac{44}{3}\right)\sqrt{2}+\left(\left(\dfrac{325}{2}\right)\sqrt{2}\right)+\dfrac{82}{7}-\left(\left(-\dfrac{86}{7}\right)\sqrt{2}\right)\right)\\ &=&\left(\dfrac{283}{105}+\left(\dfrac{4313}{60}\right)\sqrt{2}\right)-\left(\left(\dfrac{33155}{84}\right)\sqrt{2}+\dfrac{44}{21}\right)\\ &=&\dfrac{283}{105}+\left(\dfrac{4313}{60}\right)\sqrt{2}+\left(-\dfrac{33155}{84}\right)\sqrt{2}-\dfrac{44}{21}\\ &=&\dfrac{3}{5}+\left(-\dfrac{33896}{105}\right)\sqrt{2}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(5-\left(\left(2\right)\sqrt{4}\right)-\left(\left(4\right)\sqrt{18}\right)\right)-\left(\left(\left(-\dfrac{67}{5}\right)\sqrt{8}\right)-\left(\left(-\dfrac{33}{4}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{4}+\left(\dfrac{1}{5}\right)\sqrt{4}+\left(-\dfrac{59}{3}\right)\sqrt{50}\right)\right)\times\left(\left(\dfrac{65}{2}\right)\sqrt{50}-\dfrac{64}{3}-\left(\left(-2\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{4}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{4}\right)+\left(\dfrac{55}{3}\right)\sqrt{18}+\left(\dfrac{22}{3}\right)\sqrt{8}+\left(\left(\dfrac{65}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{8}\right)\right)\\ &=&\left(\left(5-4-\left(\left(12\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{134}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{165}{4}\right)\sqrt{2}\right)\right)-\dfrac{16}{3}-\left(-\dfrac{52}{7}+\dfrac{2}{5}+\left(-\dfrac{295}{3}\right)\sqrt{2}\right)\right)\times\left(\left(\dfrac{325}{2}\right)\sqrt{2}-\dfrac{64}{3}-\left(\left(-4\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{4}\right)\sqrt{2}\right)+\dfrac{82}{7}+\left(55\right)\sqrt{2}+\left(\dfrac{44}{3}\right)\sqrt{2}+\left(\left(\dfrac{325}{2}\right)\sqrt{2}\right)+\dfrac{82}{7}-\left(\left(-\dfrac{86}{7}\right)\sqrt{2}\right)\right)\\ &=&\left(\dfrac{283}{105}+\left(\dfrac{4313}{60}\right)\sqrt{2}\right)\left(\left(\dfrac{33155}{84}\right)\sqrt{2}+\dfrac{44}{21}\right)\\ &=&\left(\dfrac{1190141}{980}\right)\sqrt{2}+\dfrac{12452}{2205}+\left(\dfrac{28599503}{1008}\right)\sqrt{4}\\ &=&\left(\dfrac{1190141}{980}\right)\sqrt{2}+\dfrac{63068179923}{1111320}\\ \end{eqnarray*}