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{23}{3}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{8}{5}\right)\sqrt{20}\right)-\dfrac{25}{7}-\left(\left(-\dfrac{12}{5}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{68}{5}\right)\sqrt{125}+\left(\dfrac{67}{9}\right)\sqrt{125}+\left(\dfrac{41}{8}\right)\sqrt{125}+\left(-\dfrac{13}{2}\right)\sqrt{20}+\left(-\dfrac{21}{5}\right)\sqrt{20}\right)$ et $ Y=\left(\left(\dfrac{20}{7}\right)\sqrt{125}+\left(-\dfrac{25}{6}\right)\sqrt{45}+\left(\dfrac{55}{8}\right)\sqrt{125}+\left(-\dfrac{55}{6}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{10}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{6}{7}\right)\sqrt{45}\right)\right)-\left(\dfrac{15}{7}-\left(\left(-\dfrac{65}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{3}{7}\right)\sqrt{20}\right)\right)-\left(\left(\left(-4\right)\sqrt{25}\right)-2-\left(\left(-\dfrac{46}{3}\right)\sqrt{45}\right)\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(\left(-\dfrac{23}{3}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{8}{5}\right)\sqrt{20}\right)-\dfrac{25}{7}-\left(\left(-\dfrac{12}{5}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{68}{5}\right)\sqrt{125}+\left(\dfrac{67}{9}\right)\sqrt{125}+\left(\dfrac{41}{8}\right)\sqrt{125}+\left(-\dfrac{13}{2}\right)\sqrt{20}+\left(-\dfrac{21}{5}\right)\sqrt{20}\right)\right)+\left(\left(\left(\dfrac{20}{7}\right)\sqrt{125}+\left(-\dfrac{25}{6}\right)\sqrt{45}+\left(\dfrac{55}{8}\right)\sqrt{125}+\left(-\dfrac{55}{6}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{10}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{6}{7}\right)\sqrt{45}\right)\right)-\left(\dfrac{15}{7}-\left(\left(-\dfrac{65}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{3}{7}\right)\sqrt{20}\right)\right)-\left(\left(\left(-4\right)\sqrt{25}\right)-2-\left(\left(-\dfrac{46}{3}\right)\sqrt{45}\right)\right)\right)\\ &=&\left(\left(\left(-\dfrac{115}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{16}{5}\right)\sqrt{5}\right)-\dfrac{25}{7}-\left(\left(-\dfrac{36}{5}\right)\sqrt{5}\right)\right)-\left(\left(68\right)\sqrt{5}+\left(\dfrac{335}{9}\right)\sqrt{5}+\left(\dfrac{205}{8}\right)\sqrt{5}+\left(-13\right)\sqrt{5}+\left(-\dfrac{42}{5}\right)\sqrt{5}\right)\right)+\left(\left(\left(\dfrac{100}{7}\right)\sqrt{5}+\left(-\dfrac{25}{2}\right)\sqrt{5}+\left(\dfrac{275}{8}\right)\sqrt{5}-\dfrac{275}{6}\right)-\left(\dfrac{50}{3}-\left(\left(-\dfrac{50}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{15}{7}-\left(\left(-\dfrac{65}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{6}{7}\right)\sqrt{5}\right)\right)-\left(-20-2-\left(\left(-46\right)\sqrt{5}\right)\right)\right)\\ &=&\left(\left(-\dfrac{115}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{16}{5}\right)\sqrt{5}\right)-\dfrac{25}{7}-\left(\left(-\dfrac{36}{5}\right)\sqrt{5}\right)\right)-\left(\left(68\right)\sqrt{5}+\left(\dfrac{335}{9}\right)\sqrt{5}+\left(\dfrac{205}{8}\right)\sqrt{5}+\left(-13\right)\sqrt{5}+\left(-\dfrac{42}{5}\right)\sqrt{5}\right)+\left(\left(\dfrac{100}{7}\right)\sqrt{5}+\left(-\dfrac{25}{2}\right)\sqrt{5}+\left(\dfrac{275}{8}\right)\sqrt{5}-\dfrac{275}{6}\right)-\left(\dfrac{50}{3}-\left(\left(-\dfrac{50}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{15}{7}-\left(\left(-\dfrac{65}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{6}{7}\right)\sqrt{5}\right)\right)-\left(-20-2-\left(\left(-46\right)\sqrt{5}\right)\right)\\ &=&\left(-\dfrac{267751}{1260}\right)\sqrt{5}-\dfrac{547}{14}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(-\dfrac{23}{3}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{8}{5}\right)\sqrt{20}\right)-\dfrac{25}{7}-\left(\left(-\dfrac{12}{5}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{68}{5}\right)\sqrt{125}+\left(\dfrac{67}{9}\right)\sqrt{125}+\left(\dfrac{41}{8}\right)\sqrt{125}+\left(-\dfrac{13}{2}\right)\sqrt{20}+\left(-\dfrac{21}{5}\right)\sqrt{20}\right)\right)-\left(\left(\left(\dfrac{20}{7}\right)\sqrt{125}+\left(-\dfrac{25}{6}\right)\sqrt{45}+\left(\dfrac{55}{8}\right)\sqrt{125}+\left(-\dfrac{55}{6}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{10}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{6}{7}\right)\sqrt{45}\right)\right)-\left(\dfrac{15}{7}-\left(\left(-\dfrac{65}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{3}{7}\right)\sqrt{20}\right)\right)-\left(\left(\left(-4\right)\sqrt{25}\right)-2-\left(\left(-\dfrac{46}{3}\right)\sqrt{45}\right)\right)\right)\\ &=&\left(\left(\left(-\dfrac{115}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{16}{5}\right)\sqrt{5}\right)-\dfrac{25}{7}-\left(\left(-\dfrac{36}{5}\right)\sqrt{5}\right)\right)-\left(\left(68\right)\sqrt{5}+\left(\dfrac{335}{9}\right)\sqrt{5}+\left(\dfrac{205}{8}\right)\sqrt{5}+\left(-13\right)\sqrt{5}+\left(-\dfrac{42}{5}\right)\sqrt{5}\right)\right)-\left(\left(\left(\dfrac{100}{7}\right)\sqrt{5}+\left(-\dfrac{25}{2}\right)\sqrt{5}+\left(\dfrac{275}{8}\right)\sqrt{5}-\dfrac{275}{6}\right)-\left(\dfrac{50}{3}-\left(\left(-\dfrac{50}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{15}{7}-\left(\left(-\dfrac{65}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{6}{7}\right)\sqrt{5}\right)\right)-\left(-20-2-\left(\left(-46\right)\sqrt{5}\right)\right)\right)\\ &=&\left(\left(-\dfrac{54641}{360}\right)\sqrt{5}+\dfrac{25}{7}\right)-\left(\left(-\dfrac{10201}{168}\right)\sqrt{5}-\dfrac{597}{14}\right)\\ &=&\left(-\dfrac{54641}{360}\right)\sqrt{5}+\dfrac{25}{7}+\left(\dfrac{10201}{168}\right)\sqrt{5}+\dfrac{597}{14}\\ &=&\left(-\dfrac{28684}{315}\right)\sqrt{5}+\dfrac{647}{14}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(-\dfrac{23}{3}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{8}{5}\right)\sqrt{20}\right)-\dfrac{25}{7}-\left(\left(-\dfrac{12}{5}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{68}{5}\right)\sqrt{125}+\left(\dfrac{67}{9}\right)\sqrt{125}+\left(\dfrac{41}{8}\right)\sqrt{125}+\left(-\dfrac{13}{2}\right)\sqrt{20}+\left(-\dfrac{21}{5}\right)\sqrt{20}\right)\right)\times\left(\left(\left(\dfrac{20}{7}\right)\sqrt{125}+\left(-\dfrac{25}{6}\right)\sqrt{45}+\left(\dfrac{55}{8}\right)\sqrt{125}+\left(-\dfrac{55}{6}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{10}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{6}{7}\right)\sqrt{45}\right)\right)-\left(\dfrac{15}{7}-\left(\left(-\dfrac{65}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{3}{7}\right)\sqrt{20}\right)\right)-\left(\left(\left(-4\right)\sqrt{25}\right)-2-\left(\left(-\dfrac{46}{3}\right)\sqrt{45}\right)\right)\right)\\ &=&\left(\left(\left(-\dfrac{115}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{16}{5}\right)\sqrt{5}\right)-\dfrac{25}{7}-\left(\left(-\dfrac{36}{5}\right)\sqrt{5}\right)\right)-\left(\left(68\right)\sqrt{5}+\left(\dfrac{335}{9}\right)\sqrt{5}+\left(\dfrac{205}{8}\right)\sqrt{5}+\left(-13\right)\sqrt{5}+\left(-\dfrac{42}{5}\right)\sqrt{5}\right)\right)\times\left(\left(\left(\dfrac{100}{7}\right)\sqrt{5}+\left(-\dfrac{25}{2}\right)\sqrt{5}+\left(\dfrac{275}{8}\right)\sqrt{5}-\dfrac{275}{6}\right)-\left(\dfrac{50}{3}-\left(\left(-\dfrac{50}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{15}{7}-\left(\left(-\dfrac{65}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{6}{7}\right)\sqrt{5}\right)\right)-\left(-20-2-\left(\left(-46\right)\sqrt{5}\right)\right)\right)\\ &=&\left(\left(-\dfrac{54641}{360}\right)\sqrt{5}+\dfrac{25}{7}\right)\left(\left(-\dfrac{10201}{168}\right)\sqrt{5}-\dfrac{597}{14}\right)\\ &=&\left(\dfrac{557392841}{60480}\right)\sqrt{25}+\left(\dfrac{12358863384}{1975680}\right)\sqrt{5}-\dfrac{14925}{98}\\ &=&\dfrac{272219828090}{5927040}+\left(\dfrac{12358863384}{1975680}\right)\sqrt{5}\\ \end{eqnarray*}