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}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{46}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{27}\right)-\dfrac{79}{3}+\left(-5\right)\sqrt{12}+\left(\dfrac{79}{8}\right)\sqrt{75}+\left(-\dfrac{11}{2}\right)\sqrt{9}+\left(\left(-5\right)\sqrt{12}\right)-\left(\left(\dfrac{55}{7}\right)\sqrt{12}\right)+\left(\left(-9\right)\sqrt{75}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{12}\right)-\dfrac{25}{2}$ et $ Y=\left(\left(\dfrac{15}{7}\right)\sqrt{75}+\left(\dfrac{7}{2}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{11}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{8}\right)\sqrt{12}\right)\right)-\left(\left(\left(-3\right)\sqrt{75}\right)-\left(\left(\dfrac{79}{7}\right)\sqrt{12}\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}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{46}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{27}\right)-\dfrac{79}{3}+\left(-5\right)\sqrt{12}+\left(\dfrac{79}{8}\right)\sqrt{75}+\left(-\dfrac{11}{2}\right)\sqrt{9}+\left(\left(-5\right)\sqrt{12}\right)-\left(\left(\dfrac{55}{7}\right)\sqrt{12}\right)+\left(\left(-9\right)\sqrt{75}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{12}\right)-\dfrac{25}{2}\right)+\left(\left(\left(\dfrac{15}{7}\right)\sqrt{75}+\left(\dfrac{7}{2}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{11}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{8}\right)\sqrt{12}\right)\right)-\left(\left(\left(-3\right)\sqrt{75}\right)-\left(\left(\dfrac{79}{7}\right)\sqrt{12}\right)\right)\right)\\ &=&\left(\left(\left(\dfrac{46}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{46}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{159}{7}\right)\sqrt{3}\right)-\dfrac{79}{3}+\left(-10\right)\sqrt{3}+\left(\dfrac{395}{8}\right)\sqrt{3}-\dfrac{33}{2}+\left(\left(-10\right)\sqrt{3}\right)-\left(\left(\dfrac{110}{7}\right)\sqrt{3}\right)+\left(\left(-45\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{3}\right)-\dfrac{25}{2}\right)+\left(\left(\left(\dfrac{75}{7}\right)\sqrt{3}+\left(\dfrac{35}{2}\right)\sqrt{3}\right)-\left(\dfrac{11}{3}-\left(\left(\dfrac{21}{4}\right)\sqrt{3}\right)\right)-\left(\left(\left(-15\right)\sqrt{3}\right)-\left(\left(\dfrac{158}{7}\right)\sqrt{3}\right)\right)\right)\\ &=&\left(\left(\dfrac{46}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{46}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{159}{7}\right)\sqrt{3}\right)-\dfrac{79}{3}+\left(-10\right)\sqrt{3}+\left(\dfrac{395}{8}\right)\sqrt{3}-\dfrac{33}{2}+\left(\left(-10\right)\sqrt{3}\right)-\left(\left(\dfrac{110}{7}\right)\sqrt{3}\right)+\left(\left(-45\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{3}\right)-\dfrac{25}{2}+\left(\left(\dfrac{75}{7}\right)\sqrt{3}+\left(\dfrac{35}{2}\right)\sqrt{3}\right)-\left(\dfrac{11}{3}-\left(\left(\dfrac{21}{4}\right)\sqrt{3}\right)\right)-\left(\left(\left(-15\right)\sqrt{3}\right)-\left(\left(\dfrac{158}{7}\right)\sqrt{3}\right)\right)\\ &=&\left(\dfrac{23411}{280}\right)\sqrt{3}-59\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\dfrac{23}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{46}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{27}\right)-\dfrac{79}{3}+\left(-5\right)\sqrt{12}+\left(\dfrac{79}{8}\right)\sqrt{75}+\left(-\dfrac{11}{2}\right)\sqrt{9}+\left(\left(-5\right)\sqrt{12}\right)-\left(\left(\dfrac{55}{7}\right)\sqrt{12}\right)+\left(\left(-9\right)\sqrt{75}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{12}\right)-\dfrac{25}{2}\right)-\left(\left(\left(\dfrac{15}{7}\right)\sqrt{75}+\left(\dfrac{7}{2}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{11}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{8}\right)\sqrt{12}\right)\right)-\left(\left(\left(-3\right)\sqrt{75}\right)-\left(\left(\dfrac{79}{7}\right)\sqrt{12}\right)\right)\right)\\ &=&\left(\left(\left(\dfrac{46}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{46}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{159}{7}\right)\sqrt{3}\right)-\dfrac{79}{3}+\left(-10\right)\sqrt{3}+\left(\dfrac{395}{8}\right)\sqrt{3}-\dfrac{33}{2}+\left(\left(-10\right)\sqrt{3}\right)-\left(\left(\dfrac{110}{7}\right)\sqrt{3}\right)+\left(\left(-45\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{3}\right)-\dfrac{25}{2}\right)-\left(\left(\left(\dfrac{75}{7}\right)\sqrt{3}+\left(\dfrac{35}{2}\right)\sqrt{3}\right)-\left(\dfrac{11}{3}-\left(\left(\dfrac{21}{4}\right)\sqrt{3}\right)\right)-\left(\left(\left(-15\right)\sqrt{3}\right)-\left(\left(\dfrac{158}{7}\right)\sqrt{3}\right)\right)\right)\\ &=&\left(\left(\dfrac{503}{40}\right)\sqrt{3}-\dfrac{166}{3}\right)-\left(\left(\dfrac{1989}{28}\right)\sqrt{3}-\dfrac{11}{3}\right)\\ &=&\left(\dfrac{503}{40}\right)\sqrt{3}-\dfrac{166}{3}+\left(-\dfrac{1989}{28}\right)\sqrt{3}+\dfrac{11}{3}\\ &=&\left(-\dfrac{16369}{280}\right)\sqrt{3}-\dfrac{155}{3}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\dfrac{23}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{46}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{27}\right)-\dfrac{79}{3}+\left(-5\right)\sqrt{12}+\left(\dfrac{79}{8}\right)\sqrt{75}+\left(-\dfrac{11}{2}\right)\sqrt{9}+\left(\left(-5\right)\sqrt{12}\right)-\left(\left(\dfrac{55}{7}\right)\sqrt{12}\right)+\left(\left(-9\right)\sqrt{75}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{12}\right)-\dfrac{25}{2}\right)\times\left(\left(\left(\dfrac{15}{7}\right)\sqrt{75}+\left(\dfrac{7}{2}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{11}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{8}\right)\sqrt{12}\right)\right)-\left(\left(\left(-3\right)\sqrt{75}\right)-\left(\left(\dfrac{79}{7}\right)\sqrt{12}\right)\right)\right)\\ &=&\left(\left(\left(\dfrac{46}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{46}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{159}{7}\right)\sqrt{3}\right)-\dfrac{79}{3}+\left(-10\right)\sqrt{3}+\left(\dfrac{395}{8}\right)\sqrt{3}-\dfrac{33}{2}+\left(\left(-10\right)\sqrt{3}\right)-\left(\left(\dfrac{110}{7}\right)\sqrt{3}\right)+\left(\left(-45\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{3}\right)-\dfrac{25}{2}\right)\times\left(\left(\left(\dfrac{75}{7}\right)\sqrt{3}+\left(\dfrac{35}{2}\right)\sqrt{3}\right)-\left(\dfrac{11}{3}-\left(\left(\dfrac{21}{4}\right)\sqrt{3}\right)\right)-\left(\left(\left(-15\right)\sqrt{3}\right)-\left(\left(\dfrac{158}{7}\right)\sqrt{3}\right)\right)\right)\\ &=&\left(\left(\dfrac{503}{40}\right)\sqrt{3}-\dfrac{166}{3}\right)\left(\left(\dfrac{1989}{28}\right)\sqrt{3}-\dfrac{11}{3}\right)\\ &=&\left(\dfrac{1000467}{1120}\right)\sqrt{9}+\left(-\dfrac{3340471}{840}\right)\sqrt{3}+\dfrac{1826}{9}\\ &=&\dfrac{29057729}{10080}+\left(-\dfrac{3340471}{840}\right)\sqrt{3}\\ \end{eqnarray*}