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=-6+\left(7\right)\sqrt{45}+\left(-8\right)\sqrt{20}+\left(-3\right)\sqrt{45}+\left(5\right)\sqrt{20}+\left(1\right)\sqrt{25}+\left(-7\right)\sqrt{45}$ et $ Y=\left(\left(\left(9\right)\sqrt{25}\right)-\left(\left(\dfrac{71}{9}\right)\sqrt{125}\right)-3-\left(\left(-\dfrac{13}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{125}\right)\right)-\left(-\dfrac{77}{4}+\left(-\dfrac{19}{7}\right)\sqrt{125}+\left(\dfrac{8}{9}\right)\sqrt{25}+\left(-\dfrac{19}{3}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{45}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{53}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{125}\right)\right)-\left(\left(1\right)\sqrt{45}\right)$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(-6+\left(7\right)\sqrt{45}+\left(-8\right)\sqrt{20}+\left(-3\right)\sqrt{45}+\left(5\right)\sqrt{20}+\left(1\right)\sqrt{25}+\left(-7\right)\sqrt{45}\right)+\left(\left(\left(\left(9\right)\sqrt{25}\right)-\left(\left(\dfrac{71}{9}\right)\sqrt{125}\right)-3-\left(\left(-\dfrac{13}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{125}\right)\right)-\left(-\dfrac{77}{4}+\left(-\dfrac{19}{7}\right)\sqrt{125}+\left(\dfrac{8}{9}\right)\sqrt{25}+\left(-\dfrac{19}{3}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{45}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{53}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{125}\right)\right)-\left(\left(1\right)\sqrt{45}\right)\right)\\ &=&\left(-6+\left(21\right)\sqrt{5}+\left(-16\right)\sqrt{5}+\left(-9\right)\sqrt{5}+\left(10\right)\sqrt{5}+5+\left(-21\right)\sqrt{5}\right)+\left(\left(45-\left(\left(\dfrac{355}{9}\right)\sqrt{5}\right)-3-\left(\left(-\dfrac{13}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{215}{7}\right)\sqrt{5}\right)\right)-\left(-\dfrac{77}{4}+\left(-\dfrac{95}{7}\right)\sqrt{5}+\dfrac{40}{9}+\left(-19\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{225}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{106}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{125}{4}\right)\sqrt{5}\right)\right)-\left(\left(3\right)\sqrt{5}\right)\right)\\ &=&-6+\left(21\right)\sqrt{5}+\left(-16\right)\sqrt{5}+\left(-9\right)\sqrt{5}+\left(10\right)\sqrt{5}+5+\left(-21\right)\sqrt{5}+\left(45-\left(\left(\dfrac{355}{9}\right)\sqrt{5}\right)-3-\left(\left(-\dfrac{13}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{215}{7}\right)\sqrt{5}\right)\right)-\left(-\dfrac{77}{4}+\left(-\dfrac{95}{7}\right)\sqrt{5}+\dfrac{40}{9}+\left(-19\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{225}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{106}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{125}{4}\right)\sqrt{5}\right)\right)-\left(\left(3\right)\sqrt{5}\right)\\ &=&\dfrac{2009}{36}+\left(\dfrac{41467}{315}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(-6+\left(7\right)\sqrt{45}+\left(-8\right)\sqrt{20}+\left(-3\right)\sqrt{45}+\left(5\right)\sqrt{20}+\left(1\right)\sqrt{25}+\left(-7\right)\sqrt{45}\right)-\left(\left(\left(\left(9\right)\sqrt{25}\right)-\left(\left(\dfrac{71}{9}\right)\sqrt{125}\right)-3-\left(\left(-\dfrac{13}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{125}\right)\right)-\left(-\dfrac{77}{4}+\left(-\dfrac{19}{7}\right)\sqrt{125}+\left(\dfrac{8}{9}\right)\sqrt{25}+\left(-\dfrac{19}{3}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{45}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{53}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{125}\right)\right)-\left(\left(1\right)\sqrt{45}\right)\right)\\ &=&\left(-6+\left(21\right)\sqrt{5}+\left(-16\right)\sqrt{5}+\left(-9\right)\sqrt{5}+\left(10\right)\sqrt{5}+5+\left(-21\right)\sqrt{5}\right)-\left(\left(45-\left(\left(\dfrac{355}{9}\right)\sqrt{5}\right)-3-\left(\left(-\dfrac{13}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{215}{7}\right)\sqrt{5}\right)\right)-\left(-\dfrac{77}{4}+\left(-\dfrac{95}{7}\right)\sqrt{5}+\dfrac{40}{9}+\left(-19\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{225}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{106}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{125}{4}\right)\sqrt{5}\right)\right)-\left(\left(3\right)\sqrt{5}\right)\right)\\ &=&\left(-1+\left(-15\right)\sqrt{5}\right)-\left(\dfrac{2045}{36}+\left(\dfrac{46192}{315}\right)\sqrt{5}\right)\\ &=&-1+\left(-15\right)\sqrt{5}+-\dfrac{2045}{36}+\left(-\dfrac{46192}{315}\right)\sqrt{5}\\ &=&-\dfrac{2081}{36}+\left(-\dfrac{50917}{315}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(-6+\left(7\right)\sqrt{45}+\left(-8\right)\sqrt{20}+\left(-3\right)\sqrt{45}+\left(5\right)\sqrt{20}+\left(1\right)\sqrt{25}+\left(-7\right)\sqrt{45}\right)\times\left(\left(\left(\left(9\right)\sqrt{25}\right)-\left(\left(\dfrac{71}{9}\right)\sqrt{125}\right)-3-\left(\left(-\dfrac{13}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{125}\right)\right)-\left(-\dfrac{77}{4}+\left(-\dfrac{19}{7}\right)\sqrt{125}+\left(\dfrac{8}{9}\right)\sqrt{25}+\left(-\dfrac{19}{3}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{45}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{53}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{125}\right)\right)-\left(\left(1\right)\sqrt{45}\right)\right)\\ &=&\left(-6+\left(21\right)\sqrt{5}+\left(-16\right)\sqrt{5}+\left(-9\right)\sqrt{5}+\left(10\right)\sqrt{5}+5+\left(-21\right)\sqrt{5}\right)\times\left(\left(45-\left(\left(\dfrac{355}{9}\right)\sqrt{5}\right)-3-\left(\left(-\dfrac{13}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{215}{7}\right)\sqrt{5}\right)\right)-\left(-\dfrac{77}{4}+\left(-\dfrac{95}{7}\right)\sqrt{5}+\dfrac{40}{9}+\left(-19\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{225}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{106}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{125}{4}\right)\sqrt{5}\right)\right)-\left(\left(3\right)\sqrt{5}\right)\right)\\ &=&\left(-1+\left(-15\right)\sqrt{5}\right)\left(\dfrac{2045}{36}+\left(\dfrac{46192}{315}\right)\sqrt{5}\right)\\ &=&-\dfrac{2045}{36}+\left(-\dfrac{1258393}{1260}\right)\sqrt{5}+\left(-\dfrac{46192}{21}\right)\sqrt{25}\\ &=&-\dfrac{2785835}{252}+\left(-\dfrac{1258393}{1260}\right)\sqrt{5}\\ \end{eqnarray*}