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(-4\right)\sqrt{25}+\left(\dfrac{33}{2}\right)\sqrt{25}-\dfrac{43}{9}+\left(-7\right)\sqrt{125}+\left(\dfrac{33}{2}\right)\sqrt{25}+\left(-\dfrac{23}{3}\right)\sqrt{125}+\left(\left(-\dfrac{12}{7}\right)\sqrt{125}\right)-\left(\left(\dfrac{48}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{64}{9}\right)\sqrt{20}\right)+\left(-\dfrac{79}{7}\right)\sqrt{125}+\left(\dfrac{70}{9}\right)\sqrt{45}$ et $ Y=\left(-6\right)\sqrt{45}+\left(\left(\dfrac{65}{2}\right)\sqrt{125}\right)-\left(\left(1\right)\sqrt{20}\right)-\left(\left(\dfrac{81}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{22}{5}\right)\sqrt{20}\right)+\left(\dfrac{5}{3}\right)\sqrt{45}$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(\left(-4\right)\sqrt{25}+\left(\dfrac{33}{2}\right)\sqrt{25}-\dfrac{43}{9}+\left(-7\right)\sqrt{125}+\left(\dfrac{33}{2}\right)\sqrt{25}+\left(-\dfrac{23}{3}\right)\sqrt{125}+\left(\left(-\dfrac{12}{7}\right)\sqrt{125}\right)-\left(\left(\dfrac{48}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{64}{9}\right)\sqrt{20}\right)+\left(-\dfrac{79}{7}\right)\sqrt{125}+\left(\dfrac{70}{9}\right)\sqrt{45}\right)+\left(\left(-6\right)\sqrt{45}+\left(\left(\dfrac{65}{2}\right)\sqrt{125}\right)-\left(\left(1\right)\sqrt{20}\right)-\left(\left(\dfrac{81}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{22}{5}\right)\sqrt{20}\right)+\left(\dfrac{5}{3}\right)\sqrt{45}\right)\\ &=&\left(-20+\dfrac{165}{2}-\dfrac{43}{9}+\left(-35\right)\sqrt{5}+\dfrac{165}{2}+\left(-\dfrac{115}{3}\right)\sqrt{5}+\left(\left(-\dfrac{60}{7}\right)\sqrt{5}\right)-\left(\left(48\right)\sqrt{5}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{128}{9}\right)\sqrt{5}\right)+\left(-\dfrac{395}{7}\right)\sqrt{5}+\left(\dfrac{70}{3}\right)\sqrt{5}\right)+\left(\left(-18\right)\sqrt{5}+\left(\left(\dfrac{325}{2}\right)\sqrt{5}\right)-\left(\left(2\right)\sqrt{5}\right)-\dfrac{405}{4}-\left(\left(\dfrac{44}{5}\right)\sqrt{5}\right)+\left(5\right)\sqrt{5}\right)\\ &=&-20+\dfrac{165}{2}-\dfrac{43}{9}+\left(-35\right)\sqrt{5}+\dfrac{165}{2}+\left(-\dfrac{115}{3}\right)\sqrt{5}+\left(\left(-\dfrac{60}{7}\right)\sqrt{5}\right)-\left(\left(48\right)\sqrt{5}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{128}{9}\right)\sqrt{5}\right)+\left(-\dfrac{395}{7}\right)\sqrt{5}+\left(\dfrac{70}{3}\right)\sqrt{5}+\left(-18\right)\sqrt{5}+\left(\left(\dfrac{325}{2}\right)\sqrt{5}\right)-\left(\left(2\right)\sqrt{5}\right)-\dfrac{405}{4}-\left(\left(\dfrac{44}{5}\right)\sqrt{5}\right)+\left(5\right)\sqrt{5}\\ &=&\dfrac{1403}{36}+\left(-\dfrac{1331}{45}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(-4\right)\sqrt{25}+\left(\dfrac{33}{2}\right)\sqrt{25}-\dfrac{43}{9}+\left(-7\right)\sqrt{125}+\left(\dfrac{33}{2}\right)\sqrt{25}+\left(-\dfrac{23}{3}\right)\sqrt{125}+\left(\left(-\dfrac{12}{7}\right)\sqrt{125}\right)-\left(\left(\dfrac{48}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{64}{9}\right)\sqrt{20}\right)+\left(-\dfrac{79}{7}\right)\sqrt{125}+\left(\dfrac{70}{9}\right)\sqrt{45}\right)-\left(\left(-6\right)\sqrt{45}+\left(\left(\dfrac{65}{2}\right)\sqrt{125}\right)-\left(\left(1\right)\sqrt{20}\right)-\left(\left(\dfrac{81}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{22}{5}\right)\sqrt{20}\right)+\left(\dfrac{5}{3}\right)\sqrt{45}\right)\\ &=&\left(-20+\dfrac{165}{2}-\dfrac{43}{9}+\left(-35\right)\sqrt{5}+\dfrac{165}{2}+\left(-\dfrac{115}{3}\right)\sqrt{5}+\left(\left(-\dfrac{60}{7}\right)\sqrt{5}\right)-\left(\left(48\right)\sqrt{5}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{128}{9}\right)\sqrt{5}\right)+\left(-\dfrac{395}{7}\right)\sqrt{5}+\left(\dfrac{70}{3}\right)\sqrt{5}\right)-\left(\left(-18\right)\sqrt{5}+\left(\left(\dfrac{325}{2}\right)\sqrt{5}\right)-\left(\left(2\right)\sqrt{5}\right)-\dfrac{405}{4}-\left(\left(\dfrac{44}{5}\right)\sqrt{5}\right)+\left(5\right)\sqrt{5}\right)\\ &=&\left(\dfrac{1262}{9}+\left(-\dfrac{3029}{18}\right)\sqrt{5}\right)-\left(\left(\dfrac{1387}{10}\right)\sqrt{5}-\dfrac{405}{4}\right)\\ &=&\dfrac{1262}{9}+\left(-\dfrac{3029}{18}\right)\sqrt{5}+\left(-\dfrac{1387}{10}\right)\sqrt{5}+\dfrac{405}{4}\\ &=&\dfrac{8693}{36}+\left(-\dfrac{13814}{45}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(-4\right)\sqrt{25}+\left(\dfrac{33}{2}\right)\sqrt{25}-\dfrac{43}{9}+\left(-7\right)\sqrt{125}+\left(\dfrac{33}{2}\right)\sqrt{25}+\left(-\dfrac{23}{3}\right)\sqrt{125}+\left(\left(-\dfrac{12}{7}\right)\sqrt{125}\right)-\left(\left(\dfrac{48}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{64}{9}\right)\sqrt{20}\right)+\left(-\dfrac{79}{7}\right)\sqrt{125}+\left(\dfrac{70}{9}\right)\sqrt{45}\right)\times\left(\left(-6\right)\sqrt{45}+\left(\left(\dfrac{65}{2}\right)\sqrt{125}\right)-\left(\left(1\right)\sqrt{20}\right)-\left(\left(\dfrac{81}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{22}{5}\right)\sqrt{20}\right)+\left(\dfrac{5}{3}\right)\sqrt{45}\right)\\ &=&\left(-20+\dfrac{165}{2}-\dfrac{43}{9}+\left(-35\right)\sqrt{5}+\dfrac{165}{2}+\left(-\dfrac{115}{3}\right)\sqrt{5}+\left(\left(-\dfrac{60}{7}\right)\sqrt{5}\right)-\left(\left(48\right)\sqrt{5}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{128}{9}\right)\sqrt{5}\right)+\left(-\dfrac{395}{7}\right)\sqrt{5}+\left(\dfrac{70}{3}\right)\sqrt{5}\right)\times\left(\left(-18\right)\sqrt{5}+\left(\left(\dfrac{325}{2}\right)\sqrt{5}\right)-\left(\left(2\right)\sqrt{5}\right)-\dfrac{405}{4}-\left(\left(\dfrac{44}{5}\right)\sqrt{5}\right)+\left(5\right)\sqrt{5}\right)\\ &=&\left(\dfrac{1262}{9}+\left(-\dfrac{3029}{18}\right)\sqrt{5}\right)\left(\left(\dfrac{1387}{10}\right)\sqrt{5}-\dfrac{405}{4}\right)\\ &=&\left(\dfrac{13135301}{360}\right)\sqrt{5}-\dfrac{28395}{2}+\left(-\dfrac{4201223}{180}\right)\sqrt{25}\\ &=&\left(\dfrac{13135301}{360}\right)\sqrt{5}-\dfrac{4712333}{36}\\ \end{eqnarray*}