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=5+\dfrac{5}{2}-\left(\left(-\dfrac{56}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{125}\right)+\left(\left(\dfrac{25}{3}\right)\sqrt{125}\right)+\dfrac{43}{6}-\left(\left(-\dfrac{59}{7}\right)\sqrt{45}\right)$ et $ Y=\left(\left(-\dfrac{51}{4}\right)\sqrt{20}+\left(-\dfrac{23}{6}\right)\sqrt{20}+\left(0\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{66}{5}\right)\sqrt{125}\right)-\dfrac{48}{7}-\left(\left(-\dfrac{33}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{71}{4}\right)\sqrt{45}\right)\right)-\left(\left(\left(-\dfrac{19}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{45}{4}\right)\sqrt{20}\right)\right)-\left(-\dfrac{55}{6}-\dfrac{35}{3}-\left(\left(-5\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(5+\dfrac{5}{2}-\left(\left(-\dfrac{56}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{125}\right)+\left(\left(\dfrac{25}{3}\right)\sqrt{125}\right)+\dfrac{43}{6}-\left(\left(-\dfrac{59}{7}\right)\sqrt{45}\right)\right)+\left(\left(\left(-\dfrac{51}{4}\right)\sqrt{20}+\left(-\dfrac{23}{6}\right)\sqrt{20}+\left(0\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{66}{5}\right)\sqrt{125}\right)-\dfrac{48}{7}-\left(\left(-\dfrac{33}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{71}{4}\right)\sqrt{45}\right)\right)-\left(\left(\left(-\dfrac{19}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{45}{4}\right)\sqrt{20}\right)\right)-\left(-\dfrac{55}{6}-\dfrac{35}{3}-\left(\left(-5\right)\sqrt{125}\right)\right)\right)\\ &=&\left(5+\dfrac{5}{2}+\dfrac{280}{9}-\left(\left(-35\right)\sqrt{5}\right)-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{265}{9}\right)\sqrt{5}\right)+\left(\left(\dfrac{125}{3}\right)\sqrt{5}\right)+\dfrac{43}{6}-\left(\left(-\dfrac{177}{7}\right)\sqrt{5}\right)\right)+\left(\left(\left(-\dfrac{51}{2}\right)\sqrt{5}+\left(-\dfrac{23}{3}\right)\sqrt{5}+\left(0\right)\sqrt{5}\right)-\left(\left(\left(66\right)\sqrt{5}\right)-\dfrac{48}{7}-\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{213}{4}\right)\sqrt{5}\right)\right)-\left(-\dfrac{95}{4}-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)\right)-\left(-\dfrac{55}{6}-\dfrac{35}{3}-\left(\left(-25\right)\sqrt{5}\right)\right)\right)\\ &=&5+\dfrac{5}{2}+\dfrac{280}{9}-\left(\left(-35\right)\sqrt{5}\right)-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{265}{9}\right)\sqrt{5}\right)+\left(\left(\dfrac{125}{3}\right)\sqrt{5}\right)+\dfrac{43}{6}-\left(\left(-\dfrac{177}{7}\right)\sqrt{5}\right)+\left(\left(-\dfrac{51}{2}\right)\sqrt{5}+\left(-\dfrac{23}{3}\right)\sqrt{5}+\left(0\right)\sqrt{5}\right)-\left(\left(\left(66\right)\sqrt{5}\right)-\dfrac{48}{7}-\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{213}{4}\right)\sqrt{5}\right)\right)-\left(-\dfrac{95}{4}-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)\right)-\left(-\dfrac{55}{6}-\dfrac{35}{3}-\left(\left(-25\right)\sqrt{5}\right)\right)\\ &=&\dfrac{24499}{252}+\left(-\dfrac{7915}{252}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(5+\dfrac{5}{2}-\left(\left(-\dfrac{56}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{125}\right)+\left(\left(\dfrac{25}{3}\right)\sqrt{125}\right)+\dfrac{43}{6}-\left(\left(-\dfrac{59}{7}\right)\sqrt{45}\right)\right)-\left(\left(\left(-\dfrac{51}{4}\right)\sqrt{20}+\left(-\dfrac{23}{6}\right)\sqrt{20}+\left(0\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{66}{5}\right)\sqrt{125}\right)-\dfrac{48}{7}-\left(\left(-\dfrac{33}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{71}{4}\right)\sqrt{45}\right)\right)-\left(\left(\left(-\dfrac{19}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{45}{4}\right)\sqrt{20}\right)\right)-\left(-\dfrac{55}{6}-\dfrac{35}{3}-\left(\left(-5\right)\sqrt{125}\right)\right)\right)\\ &=&\left(5+\dfrac{5}{2}+\dfrac{280}{9}-\left(\left(-35\right)\sqrt{5}\right)-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{265}{9}\right)\sqrt{5}\right)+\left(\left(\dfrac{125}{3}\right)\sqrt{5}\right)+\dfrac{43}{6}-\left(\left(-\dfrac{177}{7}\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{51}{2}\right)\sqrt{5}+\left(-\dfrac{23}{3}\right)\sqrt{5}+\left(0\right)\sqrt{5}\right)-\left(\left(\left(66\right)\sqrt{5}\right)-\dfrac{48}{7}-\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{213}{4}\right)\sqrt{5}\right)\right)-\left(-\dfrac{95}{4}-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)\right)-\left(-\dfrac{55}{6}-\dfrac{35}{3}-\left(\left(-25\right)\sqrt{5}\right)\right)\right)\\ &=&\left(\dfrac{412}{9}+\left(\dfrac{4946}{63}\right)\sqrt{5}\right)-\left(\left(-\dfrac{1319}{12}\right)\sqrt{5}+\dfrac{4321}{84}\right)\\ &=&\dfrac{412}{9}+\left(\dfrac{4946}{63}\right)\sqrt{5}+\left(\dfrac{1319}{12}\right)\sqrt{5}-\dfrac{4321}{84}\\ &=&-\dfrac{1427}{252}+\left(\dfrac{47483}{252}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(5+\dfrac{5}{2}-\left(\left(-\dfrac{56}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{125}\right)+\left(\left(\dfrac{25}{3}\right)\sqrt{125}\right)+\dfrac{43}{6}-\left(\left(-\dfrac{59}{7}\right)\sqrt{45}\right)\right)\times\left(\left(\left(-\dfrac{51}{4}\right)\sqrt{20}+\left(-\dfrac{23}{6}\right)\sqrt{20}+\left(0\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{66}{5}\right)\sqrt{125}\right)-\dfrac{48}{7}-\left(\left(-\dfrac{33}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{71}{4}\right)\sqrt{45}\right)\right)-\left(\left(\left(-\dfrac{19}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{45}{4}\right)\sqrt{20}\right)\right)-\left(-\dfrac{55}{6}-\dfrac{35}{3}-\left(\left(-5\right)\sqrt{125}\right)\right)\right)\\ &=&\left(5+\dfrac{5}{2}+\dfrac{280}{9}-\left(\left(-35\right)\sqrt{5}\right)-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{265}{9}\right)\sqrt{5}\right)+\left(\left(\dfrac{125}{3}\right)\sqrt{5}\right)+\dfrac{43}{6}-\left(\left(-\dfrac{177}{7}\right)\sqrt{5}\right)\right)\times\left(\left(\left(-\dfrac{51}{2}\right)\sqrt{5}+\left(-\dfrac{23}{3}\right)\sqrt{5}+\left(0\right)\sqrt{5}\right)-\left(\left(\left(66\right)\sqrt{5}\right)-\dfrac{48}{7}-\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{213}{4}\right)\sqrt{5}\right)\right)-\left(-\dfrac{95}{4}-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)\right)-\left(-\dfrac{55}{6}-\dfrac{35}{3}-\left(\left(-25\right)\sqrt{5}\right)\right)\right)\\ &=&\left(\dfrac{412}{9}+\left(\dfrac{4946}{63}\right)\sqrt{5}\right)\left(\left(-\dfrac{1319}{12}\right)\sqrt{5}+\dfrac{4321}{84}\right)\\ &=&\left(-\dfrac{97339}{98}\right)\sqrt{5}+\dfrac{445063}{189}+\left(-\dfrac{3261887}{378}\right)\sqrt{25}\\ &=&\left(-\dfrac{97339}{98}\right)\sqrt{5}-\dfrac{2914249401}{71442}\\ \end{eqnarray*}