L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{25}{2}\right)\sqrt{175}-\dfrac{11}{8}+\left(-\dfrac{73}{9}\right)\sqrt{28}+\left(\dfrac{75}{8}\right)\sqrt{28}+\left(\left(\dfrac{67}{7}\right)\sqrt{49}\right)-\left(\left(-3\right)\sqrt{63}\right)-\dfrac{10}{3}-\left(\left(-\dfrac{4}{3}\right)\sqrt{49}\right)+\left(\left(\dfrac{35}{6}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{9}\right)\sqrt{63}\right)\) et \( Y=-1+\left(\dfrac{35}{3}\right)\sqrt{175}\) . 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{25}{2}\right)\sqrt{175}-\dfrac{11}{8}+\left(-\dfrac{73}{9}\right)\sqrt{28}+\left(\dfrac{75}{8}\right)\sqrt{28}+\left(\left(\dfrac{67}{7}\right)\sqrt{49}\right)-\left(\left(-3\right)\sqrt{63}\right)-\dfrac{10}{3}-\left(\left(-\dfrac{4}{3}\right)\sqrt{49}\right)+\left(\left(\dfrac{35}{6}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{9}\right)\sqrt{63}\right)\right)+\left(-1+\left(\dfrac{35}{3}\right)\sqrt{175}\right)\\
&=&\left(\left(-\dfrac{125}{2}\right)\sqrt{7}-\dfrac{11}{8}+\left(-\dfrac{146}{9}\right)\sqrt{7}+\left(\dfrac{75}{4}\right)\sqrt{7}+67-\left(\left(-9\right)\sqrt{7}\right)-\dfrac{10}{3}+\dfrac{28}{3}+\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{7}\right)\right)+\left(-1+\left(\dfrac{175}{3}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{125}{2}\right)\sqrt{7}-\dfrac{11}{8}+\left(-\dfrac{146}{9}\right)\sqrt{7}+\left(\dfrac{75}{4}\right)\sqrt{7}+67-\left(\left(-9\right)\sqrt{7}\right)-\dfrac{10}{3}+\dfrac{28}{3}+\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{7}\right)-1+\left(\dfrac{175}{3}\right)\sqrt{7}\\
&=&\left(\dfrac{667}{36}\right)\sqrt{7}+\dfrac{565}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{25}{2}\right)\sqrt{175}-\dfrac{11}{8}+\left(-\dfrac{73}{9}\right)\sqrt{28}+\left(\dfrac{75}{8}\right)\sqrt{28}+\left(\left(\dfrac{67}{7}\right)\sqrt{49}\right)-\left(\left(-3\right)\sqrt{63}\right)-\dfrac{10}{3}-\left(\left(-\dfrac{4}{3}\right)\sqrt{49}\right)+\left(\left(\dfrac{35}{6}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{9}\right)\sqrt{63}\right)\right)-\left(-1+\left(\dfrac{35}{3}\right)\sqrt{175}\right)\\
&=&\left(\left(-\dfrac{125}{2}\right)\sqrt{7}-\dfrac{11}{8}+\left(-\dfrac{146}{9}\right)\sqrt{7}+\left(\dfrac{75}{4}\right)\sqrt{7}+67-\left(\left(-9\right)\sqrt{7}\right)-\dfrac{10}{3}+\dfrac{28}{3}+\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{7}\right)\right)-\left(-1+\left(\dfrac{175}{3}\right)\sqrt{7}\right)\\
&=&\left(\left(-\dfrac{1433}{36}\right)\sqrt{7}+\dfrac{573}{8}\right)-\left(-1+\left(\dfrac{175}{3}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{1433}{36}\right)\sqrt{7}+\dfrac{573}{8}+1+\left(-\dfrac{175}{3}\right)\sqrt{7}\\
&=&\left(-\dfrac{3533}{36}\right)\sqrt{7}+\dfrac{581}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{25}{2}\right)\sqrt{175}-\dfrac{11}{8}+\left(-\dfrac{73}{9}\right)\sqrt{28}+\left(\dfrac{75}{8}\right)\sqrt{28}+\left(\left(\dfrac{67}{7}\right)\sqrt{49}\right)-\left(\left(-3\right)\sqrt{63}\right)-\dfrac{10}{3}-\left(\left(-\dfrac{4}{3}\right)\sqrt{49}\right)+\left(\left(\dfrac{35}{6}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{9}\right)\sqrt{63}\right)\right)\times\left(-1+\left(\dfrac{35}{3}\right)\sqrt{175}\right)\\
&=&\left(\left(-\dfrac{125}{2}\right)\sqrt{7}-\dfrac{11}{8}+\left(-\dfrac{146}{9}\right)\sqrt{7}+\left(\dfrac{75}{4}\right)\sqrt{7}+67-\left(\left(-9\right)\sqrt{7}\right)-\dfrac{10}{3}+\dfrac{28}{3}+\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{7}\right)\right)\times\left(-1+\left(\dfrac{175}{3}\right)\sqrt{7}\right)\\
&=&\left(\left(-\dfrac{1433}{36}\right)\sqrt{7}+\dfrac{573}{8}\right)\left(-1+\left(\dfrac{175}{3}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{303691}{72}\right)\sqrt{7}+\left(-\dfrac{250775}{108}\right)\sqrt{49}-\dfrac{573}{8}\\
&=&\left(\dfrac{303691}{72}\right)\sqrt{7}-\dfrac{3526321}{216}\\
\end{eqnarray*}