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=-\dfrac{38}{5}-\dfrac{67}{9}-\left(\left(-\dfrac{7}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{20}\right)+\left(\left(\dfrac{39}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{45}\right)+\left(\left(-\dfrac{23}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{69}{4}\right)\sqrt{45}\right)+\dfrac{32}{5}-\left(\left(-9\right)\sqrt{45}\right)-\left(\left(-\dfrac{38}{7}\right)\sqrt{125}\right)+\left(\left(1\right)\sqrt{20}\right)+\dfrac{44}{9}-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)+\left(\left(-\dfrac{27}{4}\right)\sqrt{45}\right)+\dfrac{3}{2}-\left(\left(-9\right)\sqrt{45}\right)$ et $ Y=-\dfrac{44}{7}+\left(-8\right)\sqrt{20}$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(-\dfrac{38}{5}-\dfrac{67}{9}-\left(\left(-\dfrac{7}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{20}\right)+\left(\left(\dfrac{39}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{45}\right)+\left(\left(-\dfrac{23}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{69}{4}\right)\sqrt{45}\right)+\dfrac{32}{5}-\left(\left(-9\right)\sqrt{45}\right)-\left(\left(-\dfrac{38}{7}\right)\sqrt{125}\right)+\left(\left(1\right)\sqrt{20}\right)+\dfrac{44}{9}-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)+\left(\left(-\dfrac{27}{4}\right)\sqrt{45}\right)+\dfrac{3}{2}-\left(\left(-9\right)\sqrt{45}\right)\right)+\left(-\dfrac{44}{7}+\left(-8\right)\sqrt{20}\right)\\ &=&\left(-\dfrac{38}{5}-\dfrac{67}{9}+\dfrac{35}{4}-\left(\left(-35\right)\sqrt{5}\right)+\left(\left(\dfrac{117}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{141}{8}\right)\sqrt{5}\right)+\left(\left(-\dfrac{46}{5}\right)\sqrt{5}\right)-\left(\left(-\dfrac{207}{4}\right)\sqrt{5}\right)+\dfrac{32}{5}-\left(\left(-27\right)\sqrt{5}\right)-\left(\left(-\dfrac{190}{7}\right)\sqrt{5}\right)+\left(\left(2\right)\sqrt{5}\right)+\dfrac{44}{9}-\left(\left(24\right)\sqrt{5}\right)+\left(\left(-\dfrac{81}{4}\right)\sqrt{5}\right)+\dfrac{3}{2}-\left(\left(-27\right)\sqrt{5}\right)\right)+\left(-\dfrac{44}{7}+\left(-16\right)\sqrt{5}\right)\\ &=&-\dfrac{38}{5}-\dfrac{67}{9}+\dfrac{35}{4}-\left(\left(-35\right)\sqrt{5}\right)+\left(\left(\dfrac{117}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{141}{8}\right)\sqrt{5}\right)+\left(\left(-\dfrac{46}{5}\right)\sqrt{5}\right)-\left(\left(-\dfrac{207}{4}\right)\sqrt{5}\right)+\dfrac{32}{5}-\left(\left(-27\right)\sqrt{5}\right)-\left(\left(-\dfrac{190}{7}\right)\sqrt{5}\right)+\left(\left(2\right)\sqrt{5}\right)+\dfrac{44}{9}-\left(\left(24\right)\sqrt{5}\right)+\left(\left(-\dfrac{81}{4}\right)\sqrt{5}\right)+\dfrac{3}{2}-\left(\left(-27\right)\sqrt{5}\right)-\dfrac{44}{7}+\left(-16\right)\sqrt{5}\\ &=&\dfrac{263}{1260}+\left(\dfrac{49439}{280}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(-\dfrac{38}{5}-\dfrac{67}{9}-\left(\left(-\dfrac{7}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{20}\right)+\left(\left(\dfrac{39}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{45}\right)+\left(\left(-\dfrac{23}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{69}{4}\right)\sqrt{45}\right)+\dfrac{32}{5}-\left(\left(-9\right)\sqrt{45}\right)-\left(\left(-\dfrac{38}{7}\right)\sqrt{125}\right)+\left(\left(1\right)\sqrt{20}\right)+\dfrac{44}{9}-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)+\left(\left(-\dfrac{27}{4}\right)\sqrt{45}\right)+\dfrac{3}{2}-\left(\left(-9\right)\sqrt{45}\right)\right)-\left(-\dfrac{44}{7}+\left(-8\right)\sqrt{20}\right)\\ &=&\left(-\dfrac{38}{5}-\dfrac{67}{9}+\dfrac{35}{4}-\left(\left(-35\right)\sqrt{5}\right)+\left(\left(\dfrac{117}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{141}{8}\right)\sqrt{5}\right)+\left(\left(-\dfrac{46}{5}\right)\sqrt{5}\right)-\left(\left(-\dfrac{207}{4}\right)\sqrt{5}\right)+\dfrac{32}{5}-\left(\left(-27\right)\sqrt{5}\right)-\left(\left(-\dfrac{190}{7}\right)\sqrt{5}\right)+\left(\left(2\right)\sqrt{5}\right)+\dfrac{44}{9}-\left(\left(24\right)\sqrt{5}\right)+\left(\left(-\dfrac{81}{4}\right)\sqrt{5}\right)+\dfrac{3}{2}-\left(\left(-27\right)\sqrt{5}\right)\right)-\left(-\dfrac{44}{7}+\left(-16\right)\sqrt{5}\right)\\ &=&\left(\dfrac{1169}{180}+\left(\dfrac{53919}{280}\right)\sqrt{5}\right)-\left(-\dfrac{44}{7}+\left(-16\right)\sqrt{5}\right)\\ &=&\dfrac{1169}{180}+\left(\dfrac{53919}{280}\right)\sqrt{5}+\dfrac{44}{7}+\left(16\right)\sqrt{5}\\ &=&\dfrac{16103}{1260}+\left(\dfrac{58399}{280}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(-\dfrac{38}{5}-\dfrac{67}{9}-\left(\left(-\dfrac{7}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{20}\right)+\left(\left(\dfrac{39}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{45}\right)+\left(\left(-\dfrac{23}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{69}{4}\right)\sqrt{45}\right)+\dfrac{32}{5}-\left(\left(-9\right)\sqrt{45}\right)-\left(\left(-\dfrac{38}{7}\right)\sqrt{125}\right)+\left(\left(1\right)\sqrt{20}\right)+\dfrac{44}{9}-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)+\left(\left(-\dfrac{27}{4}\right)\sqrt{45}\right)+\dfrac{3}{2}-\left(\left(-9\right)\sqrt{45}\right)\right)\times\left(-\dfrac{44}{7}+\left(-8\right)\sqrt{20}\right)\\ &=&\left(-\dfrac{38}{5}-\dfrac{67}{9}+\dfrac{35}{4}-\left(\left(-35\right)\sqrt{5}\right)+\left(\left(\dfrac{117}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{141}{8}\right)\sqrt{5}\right)+\left(\left(-\dfrac{46}{5}\right)\sqrt{5}\right)-\left(\left(-\dfrac{207}{4}\right)\sqrt{5}\right)+\dfrac{32}{5}-\left(\left(-27\right)\sqrt{5}\right)-\left(\left(-\dfrac{190}{7}\right)\sqrt{5}\right)+\left(\left(2\right)\sqrt{5}\right)+\dfrac{44}{9}-\left(\left(24\right)\sqrt{5}\right)+\left(\left(-\dfrac{81}{4}\right)\sqrt{5}\right)+\dfrac{3}{2}-\left(\left(-27\right)\sqrt{5}\right)\right)\times\left(-\dfrac{44}{7}+\left(-16\right)\sqrt{5}\right)\\ &=&\left(\dfrac{1169}{180}+\left(\dfrac{53919}{280}\right)\sqrt{5}\right)\left(-\dfrac{44}{7}+\left(-16\right)\sqrt{5}\right)\\ &=&-\dfrac{1837}{45}+\left(-\dfrac{5796229}{4410}\right)\sqrt{5}+\left(-\dfrac{107838}{35}\right)\sqrt{25}\\ &=&-\dfrac{4865569}{315}+\left(-\dfrac{5796229}{4410}\right)\sqrt{5}\\ \end{eqnarray*}