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{31}{8}\right)\sqrt{20}-\dfrac{59}{4}+\left(\dfrac{79}{5}\right)\sqrt{20}+\left(-\dfrac{25}{7}\right)\sqrt{125}+\left(\left(8\right)\sqrt{25}\right)-\left(\left(0\right)\sqrt{45}\right)-\left(\left(\dfrac{2}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{25}\right)-5+\left(\dfrac{41}{2}\right)\sqrt{125}$ et $ Y=\left(\left(\left(\dfrac{27}{4}\right)\sqrt{45}\right)+1+\dfrac{11}{3}\right)-\left(-5+\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{20}\right)-\left(-\dfrac{71}{4}+\left(\dfrac{8}{3}\right)\sqrt{20}+\left(-3\right)\sqrt{20}-\dfrac{17}{8}\right)-\left(\left(0\right)\sqrt{125}+\left(\dfrac{26}{3}\right)\sqrt{125}-\dfrac{26}{3}+\left(\dfrac{52}{7}\right)\sqrt{20}\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{31}{8}\right)\sqrt{20}-\dfrac{59}{4}+\left(\dfrac{79}{5}\right)\sqrt{20}+\left(-\dfrac{25}{7}\right)\sqrt{125}+\left(\left(8\right)\sqrt{25}\right)-\left(\left(0\right)\sqrt{45}\right)-\left(\left(\dfrac{2}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{25}\right)-5+\left(\dfrac{41}{2}\right)\sqrt{125}\right)+\left(\left(\left(\left(\dfrac{27}{4}\right)\sqrt{45}\right)+1+\dfrac{11}{3}\right)-\left(-5+\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{20}\right)-\left(-\dfrac{71}{4}+\left(\dfrac{8}{3}\right)\sqrt{20}+\left(-3\right)\sqrt{20}-\dfrac{17}{8}\right)-\left(\left(0\right)\sqrt{125}+\left(\dfrac{26}{3}\right)\sqrt{125}-\dfrac{26}{3}+\left(\dfrac{52}{7}\right)\sqrt{20}\right)\right)\\ &=&\left(\left(\dfrac{31}{4}\right)\sqrt{5}-\dfrac{59}{4}+\left(\dfrac{158}{5}\right)\sqrt{5}+\left(-\dfrac{125}{7}\right)\sqrt{5}+40-\left(\left(0\right)\sqrt{5}\right)-2+\dfrac{5}{4}-5+\left(\dfrac{205}{2}\right)\sqrt{5}\right)+\left(\left(\left(\left(\dfrac{81}{4}\right)\sqrt{5}\right)+1+\dfrac{11}{3}\right)-\left(-5+\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{156}{7}\right)\sqrt{5}\right)-\left(-\dfrac{71}{4}+\left(\dfrac{16}{3}\right)\sqrt{5}+\left(-6\right)\sqrt{5}-\dfrac{17}{8}\right)-\left(\left(0\right)\sqrt{5}+\left(\dfrac{130}{3}\right)\sqrt{5}-\dfrac{26}{3}+\left(\dfrac{104}{7}\right)\sqrt{5}\right)\right)\\ &=&\left(\dfrac{31}{4}\right)\sqrt{5}-\dfrac{59}{4}+\left(\dfrac{158}{5}\right)\sqrt{5}+\left(-\dfrac{125}{7}\right)\sqrt{5}+40-\left(\left(0\right)\sqrt{5}\right)-2+\dfrac{5}{4}-5+\left(\dfrac{205}{2}\right)\sqrt{5}+\left(\left(\left(\dfrac{81}{4}\right)\sqrt{5}\right)+1+\dfrac{11}{3}\right)-\left(-5+\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{156}{7}\right)\sqrt{5}\right)-\left(-\dfrac{71}{4}+\left(\dfrac{16}{3}\right)\sqrt{5}+\left(-6\right)\sqrt{5}-\dfrac{17}{8}\right)-\left(\left(0\right)\sqrt{5}+\left(\dfrac{130}{3}\right)\sqrt{5}-\dfrac{26}{3}+\left(\dfrac{104}{7}\right)\sqrt{5}\right)\\ &=&\left(\dfrac{533}{30}\right)\sqrt{5}+\dfrac{1385}{24}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\dfrac{31}{8}\right)\sqrt{20}-\dfrac{59}{4}+\left(\dfrac{79}{5}\right)\sqrt{20}+\left(-\dfrac{25}{7}\right)\sqrt{125}+\left(\left(8\right)\sqrt{25}\right)-\left(\left(0\right)\sqrt{45}\right)-\left(\left(\dfrac{2}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{25}\right)-5+\left(\dfrac{41}{2}\right)\sqrt{125}\right)-\left(\left(\left(\left(\dfrac{27}{4}\right)\sqrt{45}\right)+1+\dfrac{11}{3}\right)-\left(-5+\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{20}\right)-\left(-\dfrac{71}{4}+\left(\dfrac{8}{3}\right)\sqrt{20}+\left(-3\right)\sqrt{20}-\dfrac{17}{8}\right)-\left(\left(0\right)\sqrt{125}+\left(\dfrac{26}{3}\right)\sqrt{125}-\dfrac{26}{3}+\left(\dfrac{52}{7}\right)\sqrt{20}\right)\right)\\ &=&\left(\left(\dfrac{31}{4}\right)\sqrt{5}-\dfrac{59}{4}+\left(\dfrac{158}{5}\right)\sqrt{5}+\left(-\dfrac{125}{7}\right)\sqrt{5}+40-\left(\left(0\right)\sqrt{5}\right)-2+\dfrac{5}{4}-5+\left(\dfrac{205}{2}\right)\sqrt{5}\right)-\left(\left(\left(\left(\dfrac{81}{4}\right)\sqrt{5}\right)+1+\dfrac{11}{3}\right)-\left(-5+\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{156}{7}\right)\sqrt{5}\right)-\left(-\dfrac{71}{4}+\left(\dfrac{16}{3}\right)\sqrt{5}+\left(-6\right)\sqrt{5}-\dfrac{17}{8}\right)-\left(\left(0\right)\sqrt{5}+\left(\dfrac{130}{3}\right)\sqrt{5}-\dfrac{26}{3}+\left(\dfrac{104}{7}\right)\sqrt{5}\right)\right)\\ &=&\left(\left(\dfrac{17359}{140}\right)\sqrt{5}+\dfrac{39}{2}\right)-\left(\left(-\dfrac{8923}{84}\right)\sqrt{5}+\dfrac{917}{24}\right)\\ &=&\left(\dfrac{17359}{140}\right)\sqrt{5}+\dfrac{39}{2}+\left(\dfrac{8923}{84}\right)\sqrt{5}-\dfrac{917}{24}\\ &=&\left(\dfrac{24173}{105}\right)\sqrt{5}-\dfrac{449}{24}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\dfrac{31}{8}\right)\sqrt{20}-\dfrac{59}{4}+\left(\dfrac{79}{5}\right)\sqrt{20}+\left(-\dfrac{25}{7}\right)\sqrt{125}+\left(\left(8\right)\sqrt{25}\right)-\left(\left(0\right)\sqrt{45}\right)-\left(\left(\dfrac{2}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{25}\right)-5+\left(\dfrac{41}{2}\right)\sqrt{125}\right)\times\left(\left(\left(\left(\dfrac{27}{4}\right)\sqrt{45}\right)+1+\dfrac{11}{3}\right)-\left(-5+\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{20}\right)-\left(-\dfrac{71}{4}+\left(\dfrac{8}{3}\right)\sqrt{20}+\left(-3\right)\sqrt{20}-\dfrac{17}{8}\right)-\left(\left(0\right)\sqrt{125}+\left(\dfrac{26}{3}\right)\sqrt{125}-\dfrac{26}{3}+\left(\dfrac{52}{7}\right)\sqrt{20}\right)\right)\\ &=&\left(\left(\dfrac{31}{4}\right)\sqrt{5}-\dfrac{59}{4}+\left(\dfrac{158}{5}\right)\sqrt{5}+\left(-\dfrac{125}{7}\right)\sqrt{5}+40-\left(\left(0\right)\sqrt{5}\right)-2+\dfrac{5}{4}-5+\left(\dfrac{205}{2}\right)\sqrt{5}\right)\times\left(\left(\left(\left(\dfrac{81}{4}\right)\sqrt{5}\right)+1+\dfrac{11}{3}\right)-\left(-5+\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{156}{7}\right)\sqrt{5}\right)-\left(-\dfrac{71}{4}+\left(\dfrac{16}{3}\right)\sqrt{5}+\left(-6\right)\sqrt{5}-\dfrac{17}{8}\right)-\left(\left(0\right)\sqrt{5}+\left(\dfrac{130}{3}\right)\sqrt{5}-\dfrac{26}{3}+\left(\dfrac{104}{7}\right)\sqrt{5}\right)\right)\\ &=&\left(\left(\dfrac{17359}{140}\right)\sqrt{5}+\dfrac{39}{2}\right)\left(\left(-\dfrac{8923}{84}\right)\sqrt{5}+\dfrac{917}{24}\right)\\ &=&\left(-\dfrac{154894357}{11760}\right)\sqrt{25}+\left(\dfrac{8958263}{3360}\right)\sqrt{5}+\dfrac{11921}{16}\\ &=&-\dfrac{2450271520}{37632}+\left(\dfrac{8958263}{3360}\right)\sqrt{5}\\ \end{eqnarray*}