L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(4\right)\sqrt{63}\right)-\left(\left(\dfrac{23}{9}\right)\sqrt{175}\right)-\left(\left(\dfrac{19}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{67}{7}\right)\sqrt{49}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{23}{7}\right)\sqrt{28}+\left(\dfrac{35}{3}\right)\sqrt{49}+\left(-\dfrac{7}{6}\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{63}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{63}\right)-\dfrac{73}{6}+\dfrac{39}{2}\right)\) et \( Y=\left(-\dfrac{25}{4}\right)\sqrt{175}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(4\right)\sqrt{63}\right)-\left(\left(\dfrac{23}{9}\right)\sqrt{175}\right)-\left(\left(\dfrac{19}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{67}{7}\right)\sqrt{49}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{23}{7}\right)\sqrt{28}+\left(\dfrac{35}{3}\right)\sqrt{49}+\left(-\dfrac{7}{6}\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{63}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{63}\right)-\dfrac{73}{6}+\dfrac{39}{2}\right)\right)+\left(\left(-\dfrac{25}{4}\right)\sqrt{175}\right)\\
&=&\left(\left(\left(\left(12\right)\sqrt{7}\right)-\left(\left(\dfrac{115}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{95}{4}\right)\sqrt{7}\right)-67-\left(\left(-\dfrac{44}{3}\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{46}{7}\right)\sqrt{7}+\dfrac{245}{3}+\left(-\dfrac{7}{2}\right)\sqrt{7}\right)-\left(\left(\left(-28\right)\sqrt{7}\right)-\left(\left(-\dfrac{126}{5}\right)\sqrt{7}\right)-\dfrac{73}{6}+\dfrac{39}{2}\right)\right)+\left(\left(-\dfrac{125}{4}\right)\sqrt{7}\right)\\
&=&\left(\left(\left(12\right)\sqrt{7}\right)-\left(\left(\dfrac{115}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{95}{4}\right)\sqrt{7}\right)-67-\left(\left(-\dfrac{44}{3}\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{46}{7}\right)\sqrt{7}+\dfrac{245}{3}+\left(-\dfrac{7}{2}\right)\sqrt{7}\right)-\left(\left(\left(-28\right)\sqrt{7}\right)-\left(\left(-\dfrac{126}{5}\right)\sqrt{7}\right)-\dfrac{73}{6}+\dfrac{39}{2}\right)+\left(-\dfrac{125}{4}\right)\sqrt{7}\\
&=&\left(-\dfrac{17791}{630}\right)\sqrt{7}-156\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(4\right)\sqrt{63}\right)-\left(\left(\dfrac{23}{9}\right)\sqrt{175}\right)-\left(\left(\dfrac{19}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{67}{7}\right)\sqrt{49}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{23}{7}\right)\sqrt{28}+\left(\dfrac{35}{3}\right)\sqrt{49}+\left(-\dfrac{7}{6}\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{63}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{63}\right)-\dfrac{73}{6}+\dfrac{39}{2}\right)\right)-\left(\left(-\dfrac{25}{4}\right)\sqrt{175}\right)\\
&=&\left(\left(\left(\left(12\right)\sqrt{7}\right)-\left(\left(\dfrac{115}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{95}{4}\right)\sqrt{7}\right)-67-\left(\left(-\dfrac{44}{3}\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{46}{7}\right)\sqrt{7}+\dfrac{245}{3}+\left(-\dfrac{7}{2}\right)\sqrt{7}\right)-\left(\left(\left(-28\right)\sqrt{7}\right)-\left(\left(-\dfrac{126}{5}\right)\sqrt{7}\right)-\dfrac{73}{6}+\dfrac{39}{2}\right)\right)-\left(\left(-\dfrac{125}{4}\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{3793}{1260}\right)\sqrt{7}-156\right)-\left(\left(-\dfrac{125}{4}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{3793}{1260}\right)\sqrt{7}-156+\left(\dfrac{125}{4}\right)\sqrt{7}\\
&=&\left(\dfrac{10792}{315}\right)\sqrt{7}-156\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(4\right)\sqrt{63}\right)-\left(\left(\dfrac{23}{9}\right)\sqrt{175}\right)-\left(\left(\dfrac{19}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{67}{7}\right)\sqrt{49}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{23}{7}\right)\sqrt{28}+\left(\dfrac{35}{3}\right)\sqrt{49}+\left(-\dfrac{7}{6}\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{63}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{63}\right)-\dfrac{73}{6}+\dfrac{39}{2}\right)\right)\times\left(\left(-\dfrac{25}{4}\right)\sqrt{175}\right)\\
&=&\left(\left(\left(\left(12\right)\sqrt{7}\right)-\left(\left(\dfrac{115}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{95}{4}\right)\sqrt{7}\right)-67-\left(\left(-\dfrac{44}{3}\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{46}{7}\right)\sqrt{7}+\dfrac{245}{3}+\left(-\dfrac{7}{2}\right)\sqrt{7}\right)-\left(\left(\left(-28\right)\sqrt{7}\right)-\left(\left(-\dfrac{126}{5}\right)\sqrt{7}\right)-\dfrac{73}{6}+\dfrac{39}{2}\right)\right)\times\left(\left(-\dfrac{125}{4}\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{3793}{1260}\right)\sqrt{7}-156\right)\left(\left(-\dfrac{125}{4}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{94825}{1008}\right)\sqrt{49}+\left(4875\right)\sqrt{7}\\
&=&-\dfrac{94825}{144}+\left(4875\right)\sqrt{7}\\
\end{eqnarray*}