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{3}{7}\right)\sqrt{28}+\left(\dfrac{63}{4}\right)\sqrt{49}+\left(-\dfrac{25}{7}\right)\sqrt{63}+\left(0\right)\sqrt{63}\right)-\left(-\dfrac{50}{9}+\left(-\dfrac{3}{4}\right)\sqrt{28}+\left(-9\right)\sqrt{63}+\left(\dfrac{40}{9}\right)\sqrt{28}\right)-4-\left(\left(5\right)\sqrt{63}+\left(-\dfrac{36}{7}\right)\sqrt{49}+\left(5\right)\sqrt{63}+\left(0\right)\sqrt{28}\right)$ et $ Y=\left(\left(\left(\dfrac{3}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{8}{9}\right)\sqrt{175}\right)-\left(\left(3\right)\sqrt{175}\right)+4\right)-\left(\left(\dfrac{47}{4}\right)\sqrt{63}\right)-\left(\left(2\right)\sqrt{175}+\left(\dfrac{40}{9}\right)\sqrt{175}+\left(\dfrac{8}{9}\right)\sqrt{175}\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{3}{7}\right)\sqrt{28}+\left(\dfrac{63}{4}\right)\sqrt{49}+\left(-\dfrac{25}{7}\right)\sqrt{63}+\left(0\right)\sqrt{63}\right)-\left(-\dfrac{50}{9}+\left(-\dfrac{3}{4}\right)\sqrt{28}+\left(-9\right)\sqrt{63}+\left(\dfrac{40}{9}\right)\sqrt{28}\right)-4-\left(\left(5\right)\sqrt{63}+\left(-\dfrac{36}{7}\right)\sqrt{49}+\left(5\right)\sqrt{63}+\left(0\right)\sqrt{28}\right)\right)+\left(\left(\left(\left(\dfrac{3}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{8}{9}\right)\sqrt{175}\right)-\left(\left(3\right)\sqrt{175}\right)+4\right)-\left(\left(\dfrac{47}{4}\right)\sqrt{63}\right)-\left(\left(2\right)\sqrt{175}+\left(\dfrac{40}{9}\right)\sqrt{175}+\left(\dfrac{8}{9}\right)\sqrt{175}\right)\right)\\ &=&\left(\left(\left(\dfrac{6}{7}\right)\sqrt{7}+\dfrac{441}{4}+\left(-\dfrac{75}{7}\right)\sqrt{7}+\left(0\right)\sqrt{7}\right)-\left(-\dfrac{50}{9}+\left(-\dfrac{3}{2}\right)\sqrt{7}+\left(-27\right)\sqrt{7}+\left(\dfrac{80}{9}\right)\sqrt{7}\right)-4-\left(\left(15\right)\sqrt{7}-36+\left(15\right)\sqrt{7}+\left(0\right)\sqrt{7}\right)\right)+\left(\left(\dfrac{21}{2}-\left(\left(\dfrac{40}{9}\right)\sqrt{7}\right)-\left(\left(15\right)\sqrt{7}\right)+4\right)-\left(\left(\dfrac{141}{4}\right)\sqrt{7}\right)-\left(\left(10\right)\sqrt{7}+\left(\dfrac{200}{9}\right)\sqrt{7}+\left(\dfrac{40}{9}\right)\sqrt{7}\right)\right)\\ &=&\left(\left(\dfrac{6}{7}\right)\sqrt{7}+\dfrac{441}{4}+\left(-\dfrac{75}{7}\right)\sqrt{7}+\left(0\right)\sqrt{7}\right)-\left(-\dfrac{50}{9}+\left(-\dfrac{3}{2}\right)\sqrt{7}+\left(-27\right)\sqrt{7}+\left(\dfrac{80}{9}\right)\sqrt{7}\right)-4-\left(\left(15\right)\sqrt{7}-36+\left(15\right)\sqrt{7}+\left(0\right)\sqrt{7}\right)+\left(\dfrac{21}{2}-\left(\left(\dfrac{40}{9}\right)\sqrt{7}\right)-\left(\left(15\right)\sqrt{7}\right)+4\right)-\left(\left(\dfrac{141}{4}\right)\sqrt{7}\right)-\left(\left(10\right)\sqrt{7}+\left(\dfrac{200}{9}\right)\sqrt{7}+\left(\dfrac{40}{9}\right)\sqrt{7}\right)\\ &=&\left(-\dfrac{3125}{28}\right)\sqrt{7}+\dfrac{5843}{36}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\dfrac{3}{7}\right)\sqrt{28}+\left(\dfrac{63}{4}\right)\sqrt{49}+\left(-\dfrac{25}{7}\right)\sqrt{63}+\left(0\right)\sqrt{63}\right)-\left(-\dfrac{50}{9}+\left(-\dfrac{3}{4}\right)\sqrt{28}+\left(-9\right)\sqrt{63}+\left(\dfrac{40}{9}\right)\sqrt{28}\right)-4-\left(\left(5\right)\sqrt{63}+\left(-\dfrac{36}{7}\right)\sqrt{49}+\left(5\right)\sqrt{63}+\left(0\right)\sqrt{28}\right)\right)-\left(\left(\left(\left(\dfrac{3}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{8}{9}\right)\sqrt{175}\right)-\left(\left(3\right)\sqrt{175}\right)+4\right)-\left(\left(\dfrac{47}{4}\right)\sqrt{63}\right)-\left(\left(2\right)\sqrt{175}+\left(\dfrac{40}{9}\right)\sqrt{175}+\left(\dfrac{8}{9}\right)\sqrt{175}\right)\right)\\ &=&\left(\left(\left(\dfrac{6}{7}\right)\sqrt{7}+\dfrac{441}{4}+\left(-\dfrac{75}{7}\right)\sqrt{7}+\left(0\right)\sqrt{7}\right)-\left(-\dfrac{50}{9}+\left(-\dfrac{3}{2}\right)\sqrt{7}+\left(-27\right)\sqrt{7}+\left(\dfrac{80}{9}\right)\sqrt{7}\right)-4-\left(\left(15\right)\sqrt{7}-36+\left(15\right)\sqrt{7}+\left(0\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{21}{2}-\left(\left(\dfrac{40}{9}\right)\sqrt{7}\right)-\left(\left(15\right)\sqrt{7}\right)+4\right)-\left(\left(\dfrac{141}{4}\right)\sqrt{7}\right)-\left(\left(10\right)\sqrt{7}+\left(\dfrac{200}{9}\right)\sqrt{7}+\left(\dfrac{40}{9}\right)\sqrt{7}\right)\right)\\ &=&\left(\left(-\dfrac{2551}{126}\right)\sqrt{7}+\dfrac{5321}{36}\right)-\left(\dfrac{29}{2}+\left(-\dfrac{3289}{36}\right)\sqrt{7}\right)\\ &=&\left(-\dfrac{2551}{126}\right)\sqrt{7}+\dfrac{5321}{36}+-\dfrac{29}{2}+\left(\dfrac{3289}{36}\right)\sqrt{7}\\ &=&\left(\dfrac{17921}{252}\right)\sqrt{7}+\dfrac{4799}{36}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\dfrac{3}{7}\right)\sqrt{28}+\left(\dfrac{63}{4}\right)\sqrt{49}+\left(-\dfrac{25}{7}\right)\sqrt{63}+\left(0\right)\sqrt{63}\right)-\left(-\dfrac{50}{9}+\left(-\dfrac{3}{4}\right)\sqrt{28}+\left(-9\right)\sqrt{63}+\left(\dfrac{40}{9}\right)\sqrt{28}\right)-4-\left(\left(5\right)\sqrt{63}+\left(-\dfrac{36}{7}\right)\sqrt{49}+\left(5\right)\sqrt{63}+\left(0\right)\sqrt{28}\right)\right)\times\left(\left(\left(\left(\dfrac{3}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{8}{9}\right)\sqrt{175}\right)-\left(\left(3\right)\sqrt{175}\right)+4\right)-\left(\left(\dfrac{47}{4}\right)\sqrt{63}\right)-\left(\left(2\right)\sqrt{175}+\left(\dfrac{40}{9}\right)\sqrt{175}+\left(\dfrac{8}{9}\right)\sqrt{175}\right)\right)\\ &=&\left(\left(\left(\dfrac{6}{7}\right)\sqrt{7}+\dfrac{441}{4}+\left(-\dfrac{75}{7}\right)\sqrt{7}+\left(0\right)\sqrt{7}\right)-\left(-\dfrac{50}{9}+\left(-\dfrac{3}{2}\right)\sqrt{7}+\left(-27\right)\sqrt{7}+\left(\dfrac{80}{9}\right)\sqrt{7}\right)-4-\left(\left(15\right)\sqrt{7}-36+\left(15\right)\sqrt{7}+\left(0\right)\sqrt{7}\right)\right)\times\left(\left(\dfrac{21}{2}-\left(\left(\dfrac{40}{9}\right)\sqrt{7}\right)-\left(\left(15\right)\sqrt{7}\right)+4\right)-\left(\left(\dfrac{141}{4}\right)\sqrt{7}\right)-\left(\left(10\right)\sqrt{7}+\left(\dfrac{200}{9}\right)\sqrt{7}+\left(\dfrac{40}{9}\right)\sqrt{7}\right)\right)\\ &=&\left(\left(-\dfrac{2551}{126}\right)\sqrt{7}+\dfrac{5321}{36}\right)\left(\dfrac{29}{2}+\left(-\dfrac{3289}{36}\right)\sqrt{7}\right)\\ &=&\left(-\dfrac{4506070572}{326592}\right)\sqrt{7}+\left(\dfrac{8390239}{4536}\right)\sqrt{49}+\dfrac{154309}{72}\\ &=&\left(-\dfrac{4506070572}{326592}\right)\sqrt{7}+\dfrac{2444755}{162}\\ \end{eqnarray*}