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{52}{5}\right)\sqrt{175}+\left(-\dfrac{25}{3}\right)\sqrt{28}+\left(\dfrac{72}{5}\right)\sqrt{49}+\left(-7\right)\sqrt{49}+\left(\dfrac{27}{2}\right)\sqrt{28}\right)-\left(\left(\left(-2\right)\sqrt{49}\right)-\left(\left(-2\right)\sqrt{49}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{49}\right)+6-\left(\left(\dfrac{26}{3}\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{74}{5}\right)\sqrt{28}\right)-\left(\left(\left(-6\right)\sqrt{175}\right)-\left(\left(3\right)\sqrt{49}\right)\right)-\left(\left(\left(-7\right)\sqrt{49}\right)-\left(\left(7\right)\sqrt{175}\right)\right)$ et $ Y=-\dfrac{31}{3}-\left(\left(\dfrac{11}{4}\right)\sqrt{63}\right)-\left(\left(\dfrac{11}{9}\right)\sqrt{28}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{63}\right)+\left(-\dfrac{81}{5}\right)\sqrt{175}+\left(\dfrac{37}{2}\right)\sqrt{175}+\dfrac{26}{7}+\left(-\dfrac{73}{3}\right)\sqrt{28}+\left(\dfrac{8}{3}\right)\sqrt{49}+\left(-\dfrac{27}{4}\right)\sqrt{49}$ . 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{52}{5}\right)\sqrt{175}+\left(-\dfrac{25}{3}\right)\sqrt{28}+\left(\dfrac{72}{5}\right)\sqrt{49}+\left(-7\right)\sqrt{49}+\left(\dfrac{27}{2}\right)\sqrt{28}\right)-\left(\left(\left(-2\right)\sqrt{49}\right)-\left(\left(-2\right)\sqrt{49}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{49}\right)+6-\left(\left(\dfrac{26}{3}\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{74}{5}\right)\sqrt{28}\right)-\left(\left(\left(-6\right)\sqrt{175}\right)-\left(\left(3\right)\sqrt{49}\right)\right)-\left(\left(\left(-7\right)\sqrt{49}\right)-\left(\left(7\right)\sqrt{175}\right)\right)\right)+\left(-\dfrac{31}{3}-\left(\left(\dfrac{11}{4}\right)\sqrt{63}\right)-\left(\left(\dfrac{11}{9}\right)\sqrt{28}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{63}\right)+\left(-\dfrac{81}{5}\right)\sqrt{175}+\left(\dfrac{37}{2}\right)\sqrt{175}+\dfrac{26}{7}+\left(-\dfrac{73}{3}\right)\sqrt{28}+\left(\dfrac{8}{3}\right)\sqrt{49}+\left(-\dfrac{27}{4}\right)\sqrt{49}\right)\\ &=&\left(\left(\left(-52\right)\sqrt{7}+\left(-\dfrac{50}{3}\right)\sqrt{7}+\dfrac{504}{5}-49+\left(27\right)\sqrt{7}\right)-\left(-14+14-\dfrac{119}{2}+6-\left(\left(26\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{148}{5}\right)\sqrt{7}\right)-\left(\left(\left(-30\right)\sqrt{7}\right)-21\right)-\left(-49-\left(\left(35\right)\sqrt{7}\right)\right)\right)+\left(-\dfrac{31}{3}-\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{22}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)+\left(-81\right)\sqrt{7}+\left(\dfrac{185}{2}\right)\sqrt{7}+\dfrac{26}{7}+\left(-\dfrac{146}{3}\right)\sqrt{7}+\dfrac{56}{3}-\dfrac{189}{4}\right)\\ &=&\left(\left(-52\right)\sqrt{7}+\left(-\dfrac{50}{3}\right)\sqrt{7}+\dfrac{504}{5}-49+\left(27\right)\sqrt{7}\right)-\left(-14+14-\dfrac{119}{2}+6-\left(\left(26\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{148}{5}\right)\sqrt{7}\right)-\left(\left(\left(-30\right)\sqrt{7}\right)-21\right)-\left(-49-\left(\left(35\right)\sqrt{7}\right)\right)-\dfrac{31}{3}-\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{22}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)+\left(-81\right)\sqrt{7}+\left(\dfrac{185}{2}\right)\sqrt{7}+\dfrac{26}{7}+\left(-\dfrac{146}{3}\right)\sqrt{7}+\dfrac{56}{3}-\dfrac{189}{4}\\ &=&\left(\dfrac{1027}{45}\right)\sqrt{7}+\dfrac{58841}{420}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{175}+\left(-\dfrac{25}{3}\right)\sqrt{28}+\left(\dfrac{72}{5}\right)\sqrt{49}+\left(-7\right)\sqrt{49}+\left(\dfrac{27}{2}\right)\sqrt{28}\right)-\left(\left(\left(-2\right)\sqrt{49}\right)-\left(\left(-2\right)\sqrt{49}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{49}\right)+6-\left(\left(\dfrac{26}{3}\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{74}{5}\right)\sqrt{28}\right)-\left(\left(\left(-6\right)\sqrt{175}\right)-\left(\left(3\right)\sqrt{49}\right)\right)-\left(\left(\left(-7\right)\sqrt{49}\right)-\left(\left(7\right)\sqrt{175}\right)\right)\right)-\left(-\dfrac{31}{3}-\left(\left(\dfrac{11}{4}\right)\sqrt{63}\right)-\left(\left(\dfrac{11}{9}\right)\sqrt{28}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{63}\right)+\left(-\dfrac{81}{5}\right)\sqrt{175}+\left(\dfrac{37}{2}\right)\sqrt{175}+\dfrac{26}{7}+\left(-\dfrac{73}{3}\right)\sqrt{28}+\left(\dfrac{8}{3}\right)\sqrt{49}+\left(-\dfrac{27}{4}\right)\sqrt{49}\right)\\ &=&\left(\left(\left(-52\right)\sqrt{7}+\left(-\dfrac{50}{3}\right)\sqrt{7}+\dfrac{504}{5}-49+\left(27\right)\sqrt{7}\right)-\left(-14+14-\dfrac{119}{2}+6-\left(\left(26\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{148}{5}\right)\sqrt{7}\right)-\left(\left(\left(-30\right)\sqrt{7}\right)-21\right)-\left(-49-\left(\left(35\right)\sqrt{7}\right)\right)\right)-\left(-\dfrac{31}{3}-\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{22}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)+\left(-81\right)\sqrt{7}+\left(\dfrac{185}{2}\right)\sqrt{7}+\dfrac{26}{7}+\left(-\dfrac{146}{3}\right)\sqrt{7}+\dfrac{56}{3}-\dfrac{189}{4}\right)\\ &=&\left(\left(\dfrac{1184}{15}\right)\sqrt{7}+\dfrac{1753}{10}\right)-\left(-\dfrac{2957}{84}+\left(-\dfrac{505}{9}\right)\sqrt{7}\right)\\ &=&\left(\dfrac{1184}{15}\right)\sqrt{7}+\dfrac{1753}{10}+\dfrac{2957}{84}+\left(\dfrac{505}{9}\right)\sqrt{7}\\ &=&\left(\dfrac{6077}{45}\right)\sqrt{7}+\dfrac{88411}{420}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{175}+\left(-\dfrac{25}{3}\right)\sqrt{28}+\left(\dfrac{72}{5}\right)\sqrt{49}+\left(-7\right)\sqrt{49}+\left(\dfrac{27}{2}\right)\sqrt{28}\right)-\left(\left(\left(-2\right)\sqrt{49}\right)-\left(\left(-2\right)\sqrt{49}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{49}\right)+6-\left(\left(\dfrac{26}{3}\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{74}{5}\right)\sqrt{28}\right)-\left(\left(\left(-6\right)\sqrt{175}\right)-\left(\left(3\right)\sqrt{49}\right)\right)-\left(\left(\left(-7\right)\sqrt{49}\right)-\left(\left(7\right)\sqrt{175}\right)\right)\right)\times\left(-\dfrac{31}{3}-\left(\left(\dfrac{11}{4}\right)\sqrt{63}\right)-\left(\left(\dfrac{11}{9}\right)\sqrt{28}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{63}\right)+\left(-\dfrac{81}{5}\right)\sqrt{175}+\left(\dfrac{37}{2}\right)\sqrt{175}+\dfrac{26}{7}+\left(-\dfrac{73}{3}\right)\sqrt{28}+\left(\dfrac{8}{3}\right)\sqrt{49}+\left(-\dfrac{27}{4}\right)\sqrt{49}\right)\\ &=&\left(\left(\left(-52\right)\sqrt{7}+\left(-\dfrac{50}{3}\right)\sqrt{7}+\dfrac{504}{5}-49+\left(27\right)\sqrt{7}\right)-\left(-14+14-\dfrac{119}{2}+6-\left(\left(26\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{148}{5}\right)\sqrt{7}\right)-\left(\left(\left(-30\right)\sqrt{7}\right)-21\right)-\left(-49-\left(\left(35\right)\sqrt{7}\right)\right)\right)\times\left(-\dfrac{31}{3}-\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{22}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)+\left(-81\right)\sqrt{7}+\left(\dfrac{185}{2}\right)\sqrt{7}+\dfrac{26}{7}+\left(-\dfrac{146}{3}\right)\sqrt{7}+\dfrac{56}{3}-\dfrac{189}{4}\right)\\ &=&\left(\left(\dfrac{1184}{15}\right)\sqrt{7}+\dfrac{1753}{10}\right)\left(-\dfrac{2957}{84}+\left(-\dfrac{505}{9}\right)\sqrt{7}\right)\\ &=&\left(-\dfrac{2649133}{210}\right)\sqrt{7}+\left(-\dfrac{119584}{27}\right)\sqrt{49}-\dfrac{5183621}{840}\\ &=&\left(-\dfrac{2649133}{210}\right)\sqrt{7}-\dfrac{281037229}{7560}\\ \end{eqnarray*}