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{35}{4}\right)\sqrt{49}+\left(-\dfrac{79}{6}\right)\sqrt{63}+\left(-\dfrac{23}{5}\right)\sqrt{175}+\left(\dfrac{25}{2}\right)\sqrt{28}+\left(\left(\dfrac{5}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{7}\right)\sqrt{28}\right)-\left(\left(1\right)\sqrt{175}\right)+9+\left(-\dfrac{16}{3}\right)\sqrt{63}+\left(\dfrac{50}{7}\right)\sqrt{28}+\left(-\dfrac{81}{5}\right)\sqrt{63}$ et $ Y=\left(\left(\left(-\dfrac{62}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{49}\right)\right)-\left(\left(\left(-\dfrac{41}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{11}{8}\right)\sqrt{49}+\left(-\dfrac{54}{5}\right)\sqrt{175}+\dfrac{35}{3}+\left(\dfrac{71}{7}\right)\sqrt{28}\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{35}{4}\right)\sqrt{49}+\left(-\dfrac{79}{6}\right)\sqrt{63}+\left(-\dfrac{23}{5}\right)\sqrt{175}+\left(\dfrac{25}{2}\right)\sqrt{28}+\left(\left(\dfrac{5}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{7}\right)\sqrt{28}\right)-\left(\left(1\right)\sqrt{175}\right)+9+\left(-\dfrac{16}{3}\right)\sqrt{63}+\left(\dfrac{50}{7}\right)\sqrt{28}+\left(-\dfrac{81}{5}\right)\sqrt{63}\right)+\left(\left(\left(\left(-\dfrac{62}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{49}\right)\right)-\left(\left(\left(-\dfrac{41}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{11}{8}\right)\sqrt{49}+\left(-\dfrac{54}{5}\right)\sqrt{175}+\dfrac{35}{3}+\left(\dfrac{71}{7}\right)\sqrt{28}\right)\right)\\ &=&\left(-\dfrac{245}{4}+\left(-\dfrac{79}{2}\right)\sqrt{7}+\left(-23\right)\sqrt{7}+\left(25\right)\sqrt{7}+\left(\left(\dfrac{15}{4}\right)\sqrt{7}\right)+\dfrac{147}{2}-\left(\left(-\dfrac{58}{7}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)+9+\left(-16\right)\sqrt{7}+\left(\dfrac{100}{7}\right)\sqrt{7}+\left(-\dfrac{243}{5}\right)\sqrt{7}\right)+\left(\left(\left(\left(-\dfrac{310}{7}\right)\sqrt{7}\right)+\dfrac{385}{4}-28\right)-\left(-\dfrac{287}{5}-\left(\left(-13\right)\sqrt{7}\right)\right)-\left(-\dfrac{77}{8}+\left(-54\right)\sqrt{7}+\dfrac{35}{3}+\left(\dfrac{142}{7}\right)\sqrt{7}\right)\right)\\ &=&-\dfrac{245}{4}+\left(-\dfrac{79}{2}\right)\sqrt{7}+\left(-23\right)\sqrt{7}+\left(25\right)\sqrt{7}+\left(\left(\dfrac{15}{4}\right)\sqrt{7}\right)+\dfrac{147}{2}-\left(\left(-\dfrac{58}{7}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)+9+\left(-16\right)\sqrt{7}+\left(\dfrac{100}{7}\right)\sqrt{7}+\left(-\dfrac{243}{5}\right)\sqrt{7}+\left(\left(\left(-\dfrac{310}{7}\right)\sqrt{7}\right)+\dfrac{385}{4}-28\right)-\left(-\dfrac{287}{5}-\left(\left(-13\right)\sqrt{7}\right)\right)-\left(-\dfrac{77}{8}+\left(-54\right)\sqrt{7}+\dfrac{35}{3}+\left(\dfrac{142}{7}\right)\sqrt{7}\right)\\ &=&\dfrac{17383}{120}+\left(-\dfrac{2087}{20}\right)\sqrt{7}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(-\dfrac{35}{4}\right)\sqrt{49}+\left(-\dfrac{79}{6}\right)\sqrt{63}+\left(-\dfrac{23}{5}\right)\sqrt{175}+\left(\dfrac{25}{2}\right)\sqrt{28}+\left(\left(\dfrac{5}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{7}\right)\sqrt{28}\right)-\left(\left(1\right)\sqrt{175}\right)+9+\left(-\dfrac{16}{3}\right)\sqrt{63}+\left(\dfrac{50}{7}\right)\sqrt{28}+\left(-\dfrac{81}{5}\right)\sqrt{63}\right)-\left(\left(\left(\left(-\dfrac{62}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{49}\right)\right)-\left(\left(\left(-\dfrac{41}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{11}{8}\right)\sqrt{49}+\left(-\dfrac{54}{5}\right)\sqrt{175}+\dfrac{35}{3}+\left(\dfrac{71}{7}\right)\sqrt{28}\right)\right)\\ &=&\left(-\dfrac{245}{4}+\left(-\dfrac{79}{2}\right)\sqrt{7}+\left(-23\right)\sqrt{7}+\left(25\right)\sqrt{7}+\left(\left(\dfrac{15}{4}\right)\sqrt{7}\right)+\dfrac{147}{2}-\left(\left(-\dfrac{58}{7}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)+9+\left(-16\right)\sqrt{7}+\left(\dfrac{100}{7}\right)\sqrt{7}+\left(-\dfrac{243}{5}\right)\sqrt{7}\right)-\left(\left(\left(\left(-\dfrac{310}{7}\right)\sqrt{7}\right)+\dfrac{385}{4}-28\right)-\left(-\dfrac{287}{5}-\left(\left(-13\right)\sqrt{7}\right)\right)-\left(-\dfrac{77}{8}+\left(-54\right)\sqrt{7}+\dfrac{35}{3}+\left(\dfrac{142}{7}\right)\sqrt{7}\right)\right)\\ &=&\left(\dfrac{85}{4}+\left(-\dfrac{11309}{140}\right)\sqrt{7}\right)-\left(\left(-\dfrac{165}{7}\right)\sqrt{7}+\dfrac{14833}{120}\right)\\ &=&\dfrac{85}{4}+\left(-\dfrac{11309}{140}\right)\sqrt{7}+\left(\dfrac{165}{7}\right)\sqrt{7}-\dfrac{14833}{120}\\ &=&-\dfrac{12283}{120}+\left(-\dfrac{8009}{140}\right)\sqrt{7}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(-\dfrac{35}{4}\right)\sqrt{49}+\left(-\dfrac{79}{6}\right)\sqrt{63}+\left(-\dfrac{23}{5}\right)\sqrt{175}+\left(\dfrac{25}{2}\right)\sqrt{28}+\left(\left(\dfrac{5}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{7}\right)\sqrt{28}\right)-\left(\left(1\right)\sqrt{175}\right)+9+\left(-\dfrac{16}{3}\right)\sqrt{63}+\left(\dfrac{50}{7}\right)\sqrt{28}+\left(-\dfrac{81}{5}\right)\sqrt{63}\right)\times\left(\left(\left(\left(-\dfrac{62}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{49}\right)\right)-\left(\left(\left(-\dfrac{41}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{11}{8}\right)\sqrt{49}+\left(-\dfrac{54}{5}\right)\sqrt{175}+\dfrac{35}{3}+\left(\dfrac{71}{7}\right)\sqrt{28}\right)\right)\\ &=&\left(-\dfrac{245}{4}+\left(-\dfrac{79}{2}\right)\sqrt{7}+\left(-23\right)\sqrt{7}+\left(25\right)\sqrt{7}+\left(\left(\dfrac{15}{4}\right)\sqrt{7}\right)+\dfrac{147}{2}-\left(\left(-\dfrac{58}{7}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)+9+\left(-16\right)\sqrt{7}+\left(\dfrac{100}{7}\right)\sqrt{7}+\left(-\dfrac{243}{5}\right)\sqrt{7}\right)\times\left(\left(\left(\left(-\dfrac{310}{7}\right)\sqrt{7}\right)+\dfrac{385}{4}-28\right)-\left(-\dfrac{287}{5}-\left(\left(-13\right)\sqrt{7}\right)\right)-\left(-\dfrac{77}{8}+\left(-54\right)\sqrt{7}+\dfrac{35}{3}+\left(\dfrac{142}{7}\right)\sqrt{7}\right)\right)\\ &=&\left(\dfrac{85}{4}+\left(-\dfrac{11309}{140}\right)\sqrt{7}\right)\left(\left(-\dfrac{165}{7}\right)\sqrt{7}+\dfrac{14833}{120}\right)\\ &=&\left(-\dfrac{176161397}{16800}\right)\sqrt{7}+\dfrac{252161}{96}+\left(\dfrac{373197}{196}\right)\sqrt{49}\\ &=&\left(-\dfrac{176161397}{16800}\right)\sqrt{7}+\dfrac{10721855}{672}\\ \end{eqnarray*}