L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{5}{3}-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(\dfrac{8}{7}\right)\sqrt{27}\right)-\dfrac{9}{4}\right)-\left(\left(\left(9\right)\sqrt{9}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(5\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{38}{5}\right)\sqrt{27}\right)\) et \( Y=\left(\left(-\dfrac{22}{5}\right)\sqrt{27}-\dfrac{70}{3}\right)-\left(\left(-\dfrac{25}{9}\right)\sqrt{12}+\left(\dfrac{14}{3}\right)\sqrt{75}+\left(\dfrac{17}{4}\right)\sqrt{75}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{5}{3}-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(\dfrac{8}{7}\right)\sqrt{27}\right)-\dfrac{9}{4}\right)-\left(\left(\left(9\right)\sqrt{9}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(5\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{38}{5}\right)\sqrt{27}\right)\right)+\left(\left(\left(-\dfrac{22}{5}\right)\sqrt{27}-\dfrac{70}{3}\right)-\left(\left(-\dfrac{25}{9}\right)\sqrt{12}+\left(\dfrac{14}{3}\right)\sqrt{75}+\left(\dfrac{17}{4}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-\dfrac{5}{3}+24-\left(\left(\dfrac{24}{7}\right)\sqrt{3}\right)-\dfrac{9}{4}\right)-\left(27-\left(\left(-\dfrac{162}{7}\right)\sqrt{3}\right)+24-\left(\left(25\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{114}{5}\right)\sqrt{3}\right)\right)+\left(\left(\left(-\dfrac{66}{5}\right)\sqrt{3}-\dfrac{70}{3}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{3}+\left(\dfrac{70}{3}\right)\sqrt{3}+\left(\dfrac{85}{4}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{5}{3}+24-\left(\left(\dfrac{24}{7}\right)\sqrt{3}\right)-\dfrac{9}{4}\right)-\left(27-\left(\left(-\dfrac{162}{7}\right)\sqrt{3}\right)+24-\left(\left(25\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{114}{5}\right)\sqrt{3}\right)+\left(\left(-\dfrac{66}{5}\right)\sqrt{3}-\dfrac{70}{3}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{3}+\left(\dfrac{70}{3}\right)\sqrt{3}+\left(\dfrac{85}{4}\right)\sqrt{3}\right)\\
&=&-\dfrac{217}{4}+\left(-\dfrac{19303}{252}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{5}{3}-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(\dfrac{8}{7}\right)\sqrt{27}\right)-\dfrac{9}{4}\right)-\left(\left(\left(9\right)\sqrt{9}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(5\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{38}{5}\right)\sqrt{27}\right)\right)-\left(\left(\left(-\dfrac{22}{5}\right)\sqrt{27}-\dfrac{70}{3}\right)-\left(\left(-\dfrac{25}{9}\right)\sqrt{12}+\left(\dfrac{14}{3}\right)\sqrt{75}+\left(\dfrac{17}{4}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-\dfrac{5}{3}+24-\left(\left(\dfrac{24}{7}\right)\sqrt{3}\right)-\dfrac{9}{4}\right)-\left(27-\left(\left(-\dfrac{162}{7}\right)\sqrt{3}\right)+24-\left(\left(25\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{114}{5}\right)\sqrt{3}\right)\right)-\left(\left(\left(-\dfrac{66}{5}\right)\sqrt{3}-\dfrac{70}{3}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{3}+\left(\dfrac{70}{3}\right)\sqrt{3}+\left(\dfrac{85}{4}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{371}{12}+\left(-\dfrac{853}{35}\right)\sqrt{3}\right)-\left(\left(-\dfrac{9401}{180}\right)\sqrt{3}-\dfrac{70}{3}\right)\\
&=&-\dfrac{371}{12}+\left(-\dfrac{853}{35}\right)\sqrt{3}+\left(\dfrac{9401}{180}\right)\sqrt{3}+\dfrac{70}{3}\\
&=&-\dfrac{91}{12}+\left(\dfrac{35099}{1260}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{5}{3}-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(\dfrac{8}{7}\right)\sqrt{27}\right)-\dfrac{9}{4}\right)-\left(\left(\left(9\right)\sqrt{9}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(5\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{38}{5}\right)\sqrt{27}\right)\right)\times\left(\left(\left(-\dfrac{22}{5}\right)\sqrt{27}-\dfrac{70}{3}\right)-\left(\left(-\dfrac{25}{9}\right)\sqrt{12}+\left(\dfrac{14}{3}\right)\sqrt{75}+\left(\dfrac{17}{4}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-\dfrac{5}{3}+24-\left(\left(\dfrac{24}{7}\right)\sqrt{3}\right)-\dfrac{9}{4}\right)-\left(27-\left(\left(-\dfrac{162}{7}\right)\sqrt{3}\right)+24-\left(\left(25\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{114}{5}\right)\sqrt{3}\right)\right)\times\left(\left(\left(-\dfrac{66}{5}\right)\sqrt{3}-\dfrac{70}{3}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{3}+\left(\dfrac{70}{3}\right)\sqrt{3}+\left(\dfrac{85}{4}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{371}{12}+\left(-\dfrac{853}{35}\right)\sqrt{3}\right)\left(\left(-\dfrac{9401}{180}\right)\sqrt{3}-\dfrac{70}{3}\right)\\
&=&\left(\dfrac{4716091}{2160}\right)\sqrt{3}+\dfrac{12985}{18}+\left(\dfrac{1145579}{900}\right)\sqrt{9}\\
&=&\left(\dfrac{4716091}{2160}\right)\sqrt{3}+\dfrac{4085987}{900}\\
\end{eqnarray*}