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(\left(\dfrac{75}{4}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{75}\right)+5\right)-\left(\dfrac{13}{4}+\left(\dfrac{3}{2}\right)\sqrt{27}+\left(-\dfrac{19}{9}\right)\sqrt{75}+\left(\dfrac{17}{9}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{19}{9}\right)\sqrt{75}\right)-\left(\left(2\right)\sqrt{75}\right)\right)$ et $ Y=\left(7-\dfrac{23}{3}\right)-\left(\dfrac{79}{5}-\left(\left(-\dfrac{31}{3}\right)\sqrt{12}\right)\right)-\left(\left(-7\right)\sqrt{27}\right)+6-\left(\left(\left(-\dfrac{13}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)-\dfrac{2}{3}\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(\left(\dfrac{75}{4}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{75}\right)+5\right)-\left(\dfrac{13}{4}+\left(\dfrac{3}{2}\right)\sqrt{27}+\left(-\dfrac{19}{9}\right)\sqrt{75}+\left(\dfrac{17}{9}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{19}{9}\right)\sqrt{75}\right)-\left(\left(2\right)\sqrt{75}\right)\right)\right)+\left(\left(7-\dfrac{23}{3}\right)-\left(\dfrac{79}{5}-\left(\left(-\dfrac{31}{3}\right)\sqrt{12}\right)\right)-\left(\left(-7\right)\sqrt{27}\right)+6-\left(\left(\left(-\dfrac{13}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)-\dfrac{2}{3}\right)\right)\\ &=&\left(\left(\left(\dfrac{375}{4}\right)\sqrt{3}\right)-\left(-\dfrac{153}{8}-\left(\left(35\right)\sqrt{3}\right)+5\right)-\left(\dfrac{13}{4}+\left(\dfrac{9}{2}\right)\sqrt{3}+\left(-\dfrac{95}{9}\right)\sqrt{3}+\left(\dfrac{34}{9}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{95}{9}\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)\right)\right)+\left(\left(7-\dfrac{23}{3}\right)-\left(\dfrac{79}{5}-\left(\left(-\dfrac{62}{3}\right)\sqrt{3}\right)\right)-\left(\left(-21\right)\sqrt{3}\right)+6-\left(-\dfrac{39}{5}-\left(\left(-\dfrac{50}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{57}{4}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\dfrac{2}{3}\right)\right)\\ &=&\left(\left(\dfrac{375}{4}\right)\sqrt{3}\right)-\left(-\dfrac{153}{8}-\left(\left(35\right)\sqrt{3}\right)+5\right)-\left(\dfrac{13}{4}+\left(\dfrac{9}{2}\right)\sqrt{3}+\left(-\dfrac{95}{9}\right)\sqrt{3}+\left(\dfrac{34}{9}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{95}{9}\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)\right)+\left(7-\dfrac{23}{3}\right)-\left(\dfrac{79}{5}-\left(\left(-\dfrac{62}{3}\right)\sqrt{3}\right)\right)-\left(\left(-21\right)\sqrt{3}\right)+6-\left(-\dfrac{39}{5}-\left(\left(-\dfrac{50}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{57}{4}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\dfrac{2}{3}\right)\\ &=&\left(100\right)\sqrt{3}+\dfrac{71}{8}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\dfrac{75}{4}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{75}\right)+5\right)-\left(\dfrac{13}{4}+\left(\dfrac{3}{2}\right)\sqrt{27}+\left(-\dfrac{19}{9}\right)\sqrt{75}+\left(\dfrac{17}{9}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{19}{9}\right)\sqrt{75}\right)-\left(\left(2\right)\sqrt{75}\right)\right)\right)-\left(\left(7-\dfrac{23}{3}\right)-\left(\dfrac{79}{5}-\left(\left(-\dfrac{31}{3}\right)\sqrt{12}\right)\right)-\left(\left(-7\right)\sqrt{27}\right)+6-\left(\left(\left(-\dfrac{13}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)-\dfrac{2}{3}\right)\right)\\ &=&\left(\left(\left(\dfrac{375}{4}\right)\sqrt{3}\right)-\left(-\dfrac{153}{8}-\left(\left(35\right)\sqrt{3}\right)+5\right)-\left(\dfrac{13}{4}+\left(\dfrac{9}{2}\right)\sqrt{3}+\left(-\dfrac{95}{9}\right)\sqrt{3}+\left(\dfrac{34}{9}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{95}{9}\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)\right)\right)-\left(\left(7-\dfrac{23}{3}\right)-\left(\dfrac{79}{5}-\left(\left(-\dfrac{62}{3}\right)\sqrt{3}\right)\right)-\left(\left(-21\right)\sqrt{3}\right)+6-\left(-\dfrac{39}{5}-\left(\left(-\dfrac{50}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{57}{4}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\dfrac{2}{3}\right)\right)\\ &=&\left(\left(\dfrac{1819}{12}\right)\sqrt{3}+\dfrac{87}{8}\right)-\left(-2+\left(-\dfrac{619}{12}\right)\sqrt{3}\right)\\ &=&\left(\dfrac{1819}{12}\right)\sqrt{3}+\dfrac{87}{8}+2+\left(\dfrac{619}{12}\right)\sqrt{3}\\ &=&\left(\dfrac{1219}{6}\right)\sqrt{3}+\dfrac{103}{8}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\dfrac{75}{4}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{75}\right)+5\right)-\left(\dfrac{13}{4}+\left(\dfrac{3}{2}\right)\sqrt{27}+\left(-\dfrac{19}{9}\right)\sqrt{75}+\left(\dfrac{17}{9}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{19}{9}\right)\sqrt{75}\right)-\left(\left(2\right)\sqrt{75}\right)\right)\right)\times\left(\left(7-\dfrac{23}{3}\right)-\left(\dfrac{79}{5}-\left(\left(-\dfrac{31}{3}\right)\sqrt{12}\right)\right)-\left(\left(-7\right)\sqrt{27}\right)+6-\left(\left(\left(-\dfrac{13}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)-\dfrac{2}{3}\right)\right)\\ &=&\left(\left(\left(\dfrac{375}{4}\right)\sqrt{3}\right)-\left(-\dfrac{153}{8}-\left(\left(35\right)\sqrt{3}\right)+5\right)-\left(\dfrac{13}{4}+\left(\dfrac{9}{2}\right)\sqrt{3}+\left(-\dfrac{95}{9}\right)\sqrt{3}+\left(\dfrac{34}{9}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{95}{9}\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)\right)\right)\times\left(\left(7-\dfrac{23}{3}\right)-\left(\dfrac{79}{5}-\left(\left(-\dfrac{62}{3}\right)\sqrt{3}\right)\right)-\left(\left(-21\right)\sqrt{3}\right)+6-\left(-\dfrac{39}{5}-\left(\left(-\dfrac{50}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{57}{4}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\dfrac{2}{3}\right)\right)\\ &=&\left(\left(\dfrac{1819}{12}\right)\sqrt{3}+\dfrac{87}{8}\right)\left(-2+\left(-\dfrac{619}{12}\right)\sqrt{3}\right)\\ &=&\left(-\dfrac{82957}{96}\right)\sqrt{3}+\left(-\dfrac{1125961}{144}\right)\sqrt{9}-\dfrac{87}{4}\\ &=&\left(-\dfrac{82957}{96}\right)\sqrt{3}-\dfrac{1127005}{48}\\ \end{eqnarray*}