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{5}{4}\right)\sqrt{20}+\left(\dfrac{25}{3}\right)\sqrt{20}+\left(-2\right)\sqrt{125}+\dfrac{23}{7}+\left(-\dfrac{58}{3}\right)\sqrt{45}+\left(\dfrac{52}{7}\right)\sqrt{25}-\dfrac{1}{2}-\dfrac{77}{4}+\left(-\dfrac{55}{4}\right)\sqrt{125}+\left(\dfrac{74}{5}\right)\sqrt{45}+\left(0\right)\sqrt{125}+\left(\left(-\dfrac{41}{8}\right)\sqrt{25}\right)+\dfrac{65}{6}-\left(\left(6\right)\sqrt{45}\right)+\left(\dfrac{65}{2}\right)\sqrt{125}$ et $ Y=\left(-\dfrac{17}{2}\right)\sqrt{20}+\left(\dfrac{39}{5}\right)\sqrt{25}+\left(\left(\dfrac{13}{7}\right)\sqrt{20}\right)-\left(\left(5\right)\sqrt{125}\right)-\left(\left(\dfrac{57}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{49}{2}\right)\sqrt{20}\right)+\left(\dfrac{7}{2}\right)\sqrt{25}+\dfrac{23}{5}-1+\left(-\dfrac{64}{7}\right)\sqrt{25}+\left(-\dfrac{2}{3}\right)\sqrt{45}+\left(\left(-\dfrac{54}{5}\right)\sqrt{25}\right)-\left(\left(-7\right)\sqrt{125}\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{5}{4}\right)\sqrt{20}+\left(\dfrac{25}{3}\right)\sqrt{20}+\left(-2\right)\sqrt{125}+\dfrac{23}{7}+\left(-\dfrac{58}{3}\right)\sqrt{45}+\left(\dfrac{52}{7}\right)\sqrt{25}-\dfrac{1}{2}-\dfrac{77}{4}+\left(-\dfrac{55}{4}\right)\sqrt{125}+\left(\dfrac{74}{5}\right)\sqrt{45}+\left(0\right)\sqrt{125}+\left(\left(-\dfrac{41}{8}\right)\sqrt{25}\right)+\dfrac{65}{6}-\left(\left(6\right)\sqrt{45}\right)+\left(\dfrac{65}{2}\right)\sqrt{125}\right)+\left(\left(-\dfrac{17}{2}\right)\sqrt{20}+\left(\dfrac{39}{5}\right)\sqrt{25}+\left(\left(\dfrac{13}{7}\right)\sqrt{20}\right)-\left(\left(5\right)\sqrt{125}\right)-\left(\left(\dfrac{57}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{49}{2}\right)\sqrt{20}\right)+\left(\dfrac{7}{2}\right)\sqrt{25}+\dfrac{23}{5}-1+\left(-\dfrac{64}{7}\right)\sqrt{25}+\left(-\dfrac{2}{3}\right)\sqrt{45}+\left(\left(-\dfrac{54}{5}\right)\sqrt{25}\right)-\left(\left(-7\right)\sqrt{125}\right)\right)\\ &=&\left(\left(-\dfrac{5}{2}\right)\sqrt{5}+\left(\dfrac{50}{3}\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\dfrac{23}{7}+\left(-58\right)\sqrt{5}+\dfrac{260}{7}-\dfrac{1}{2}-\dfrac{77}{4}+\left(-\dfrac{275}{4}\right)\sqrt{5}+\left(\dfrac{222}{5}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{205}{8}+\dfrac{65}{6}-\left(\left(18\right)\sqrt{5}\right)+\left(\dfrac{325}{2}\right)\sqrt{5}\right)+\left(\left(-17\right)\sqrt{5}+39+\left(\left(\dfrac{26}{7}\right)\sqrt{5}\right)-\left(\left(25\right)\sqrt{5}\right)-\left(\left(\dfrac{114}{7}\right)\sqrt{5}\right)-\left(\left(49\right)\sqrt{5}\right)+\dfrac{35}{2}+\dfrac{23}{5}-1-\dfrac{320}{7}+\left(-2\right)\sqrt{5}-54-\left(\left(-35\right)\sqrt{5}\right)\right)\\ &=&\left(-\dfrac{5}{2}\right)\sqrt{5}+\left(\dfrac{50}{3}\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\dfrac{23}{7}+\left(-58\right)\sqrt{5}+\dfrac{260}{7}-\dfrac{1}{2}-\dfrac{77}{4}+\left(-\dfrac{275}{4}\right)\sqrt{5}+\left(\dfrac{222}{5}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{205}{8}+\dfrac{65}{6}-\left(\left(18\right)\sqrt{5}\right)+\left(\dfrac{325}{2}\right)\sqrt{5}+\left(-17\right)\sqrt{5}+39+\left(\left(\dfrac{26}{7}\right)\sqrt{5}\right)-\left(\left(25\right)\sqrt{5}\right)-\left(\left(\dfrac{114}{7}\right)\sqrt{5}\right)-\left(\left(49\right)\sqrt{5}\right)+\dfrac{35}{2}+\dfrac{23}{5}-1-\dfrac{320}{7}+\left(-2\right)\sqrt{5}-54-\left(\left(-35\right)\sqrt{5}\right)\\ &=&\left(-\dfrac{1787}{420}\right)\sqrt{5}-\dfrac{28331}{840}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(-\dfrac{5}{4}\right)\sqrt{20}+\left(\dfrac{25}{3}\right)\sqrt{20}+\left(-2\right)\sqrt{125}+\dfrac{23}{7}+\left(-\dfrac{58}{3}\right)\sqrt{45}+\left(\dfrac{52}{7}\right)\sqrt{25}-\dfrac{1}{2}-\dfrac{77}{4}+\left(-\dfrac{55}{4}\right)\sqrt{125}+\left(\dfrac{74}{5}\right)\sqrt{45}+\left(0\right)\sqrt{125}+\left(\left(-\dfrac{41}{8}\right)\sqrt{25}\right)+\dfrac{65}{6}-\left(\left(6\right)\sqrt{45}\right)+\left(\dfrac{65}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{20}+\left(\dfrac{39}{5}\right)\sqrt{25}+\left(\left(\dfrac{13}{7}\right)\sqrt{20}\right)-\left(\left(5\right)\sqrt{125}\right)-\left(\left(\dfrac{57}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{49}{2}\right)\sqrt{20}\right)+\left(\dfrac{7}{2}\right)\sqrt{25}+\dfrac{23}{5}-1+\left(-\dfrac{64}{7}\right)\sqrt{25}+\left(-\dfrac{2}{3}\right)\sqrt{45}+\left(\left(-\dfrac{54}{5}\right)\sqrt{25}\right)-\left(\left(-7\right)\sqrt{125}\right)\right)\\ &=&\left(\left(-\dfrac{5}{2}\right)\sqrt{5}+\left(\dfrac{50}{3}\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\dfrac{23}{7}+\left(-58\right)\sqrt{5}+\dfrac{260}{7}-\dfrac{1}{2}-\dfrac{77}{4}+\left(-\dfrac{275}{4}\right)\sqrt{5}+\left(\dfrac{222}{5}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{205}{8}+\dfrac{65}{6}-\left(\left(18\right)\sqrt{5}\right)+\left(\dfrac{325}{2}\right)\sqrt{5}\right)-\left(\left(-17\right)\sqrt{5}+39+\left(\left(\dfrac{26}{7}\right)\sqrt{5}\right)-\left(\left(25\right)\sqrt{5}\right)-\left(\left(\dfrac{114}{7}\right)\sqrt{5}\right)-\left(\left(49\right)\sqrt{5}\right)+\dfrac{35}{2}+\dfrac{23}{5}-1-\dfrac{320}{7}+\left(-2\right)\sqrt{5}-54-\left(\left(-35\right)\sqrt{5}\right)\right)\\ &=&\left(\left(\dfrac{3979}{60}\right)\sqrt{5}+\dfrac{989}{168}\right)-\left(\left(-\dfrac{494}{7}\right)\sqrt{5}-\dfrac{2773}{70}\right)\\ &=&\left(\dfrac{3979}{60}\right)\sqrt{5}+\dfrac{989}{168}+\left(\dfrac{494}{7}\right)\sqrt{5}+\dfrac{2773}{70}\\ &=&\left(\dfrac{57493}{420}\right)\sqrt{5}+\dfrac{38221}{840}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(-\dfrac{5}{4}\right)\sqrt{20}+\left(\dfrac{25}{3}\right)\sqrt{20}+\left(-2\right)\sqrt{125}+\dfrac{23}{7}+\left(-\dfrac{58}{3}\right)\sqrt{45}+\left(\dfrac{52}{7}\right)\sqrt{25}-\dfrac{1}{2}-\dfrac{77}{4}+\left(-\dfrac{55}{4}\right)\sqrt{125}+\left(\dfrac{74}{5}\right)\sqrt{45}+\left(0\right)\sqrt{125}+\left(\left(-\dfrac{41}{8}\right)\sqrt{25}\right)+\dfrac{65}{6}-\left(\left(6\right)\sqrt{45}\right)+\left(\dfrac{65}{2}\right)\sqrt{125}\right)\times\left(\left(-\dfrac{17}{2}\right)\sqrt{20}+\left(\dfrac{39}{5}\right)\sqrt{25}+\left(\left(\dfrac{13}{7}\right)\sqrt{20}\right)-\left(\left(5\right)\sqrt{125}\right)-\left(\left(\dfrac{57}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{49}{2}\right)\sqrt{20}\right)+\left(\dfrac{7}{2}\right)\sqrt{25}+\dfrac{23}{5}-1+\left(-\dfrac{64}{7}\right)\sqrt{25}+\left(-\dfrac{2}{3}\right)\sqrt{45}+\left(\left(-\dfrac{54}{5}\right)\sqrt{25}\right)-\left(\left(-7\right)\sqrt{125}\right)\right)\\ &=&\left(\left(-\dfrac{5}{2}\right)\sqrt{5}+\left(\dfrac{50}{3}\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\dfrac{23}{7}+\left(-58\right)\sqrt{5}+\dfrac{260}{7}-\dfrac{1}{2}-\dfrac{77}{4}+\left(-\dfrac{275}{4}\right)\sqrt{5}+\left(\dfrac{222}{5}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{205}{8}+\dfrac{65}{6}-\left(\left(18\right)\sqrt{5}\right)+\left(\dfrac{325}{2}\right)\sqrt{5}\right)\times\left(\left(-17\right)\sqrt{5}+39+\left(\left(\dfrac{26}{7}\right)\sqrt{5}\right)-\left(\left(25\right)\sqrt{5}\right)-\left(\left(\dfrac{114}{7}\right)\sqrt{5}\right)-\left(\left(49\right)\sqrt{5}\right)+\dfrac{35}{2}+\dfrac{23}{5}-1-\dfrac{320}{7}+\left(-2\right)\sqrt{5}-54-\left(\left(-35\right)\sqrt{5}\right)\right)\\ &=&\left(\left(\dfrac{3979}{60}\right)\sqrt{5}+\dfrac{989}{168}\right)\left(\left(-\dfrac{494}{7}\right)\sqrt{5}-\dfrac{2773}{70}\right)\\ &=&\left(-\dfrac{982813}{210}\right)\sqrt{25}+\left(-\dfrac{7513843596}{2469600}\right)\sqrt{5}-\dfrac{2742497}{11760}\\ &=&-\dfrac{11673065754}{493920}+\left(-\dfrac{7513843596}{2469600}\right)\sqrt{5}\\ \end{eqnarray*}