L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{21}{4}\right)\sqrt{63}\) et \( Y=\left(\left(\dfrac{41}{5}\right)\sqrt{63}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{43}{8}\right)\sqrt{175}\right)+\left(7\right)\sqrt{28}\) . 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{21}{4}\right)\sqrt{63}\right)+\left(\left(\left(\dfrac{41}{5}\right)\sqrt{63}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{43}{8}\right)\sqrt{175}\right)+\left(7\right)\sqrt{28}\right)\\
&=&\left(\left(\dfrac{63}{4}\right)\sqrt{7}\right)+\left(\left(\left(\dfrac{123}{5}\right)\sqrt{7}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{215}{8}\right)\sqrt{7}\right)+\left(14\right)\sqrt{7}\right)\\
&=&\left(\dfrac{63}{4}\right)\sqrt{7}+\left(\left(\dfrac{123}{5}\right)\sqrt{7}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{215}{8}\right)\sqrt{7}\right)+\left(14\right)\sqrt{7}\\
&=&\left(\dfrac{3249}{40}\right)\sqrt{7}+\dfrac{44}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{21}{4}\right)\sqrt{63}\right)-\left(\left(\left(\dfrac{41}{5}\right)\sqrt{63}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{43}{8}\right)\sqrt{175}\right)+\left(7\right)\sqrt{28}\right)\\
&=&\left(\left(\dfrac{63}{4}\right)\sqrt{7}\right)-\left(\left(\left(\dfrac{123}{5}\right)\sqrt{7}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{215}{8}\right)\sqrt{7}\right)+\left(14\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{63}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{2619}{40}\right)\sqrt{7}+\dfrac{44}{7}\right)\\
&=&\left(\dfrac{63}{4}\right)\sqrt{7}+\left(-\dfrac{2619}{40}\right)\sqrt{7}-\dfrac{44}{7}\\
&=&\left(-\dfrac{1989}{40}\right)\sqrt{7}-\dfrac{44}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{21}{4}\right)\sqrt{63}\right)\times\left(\left(\left(\dfrac{41}{5}\right)\sqrt{63}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{43}{8}\right)\sqrt{175}\right)+\left(7\right)\sqrt{28}\right)\\
&=&\left(\left(\dfrac{63}{4}\right)\sqrt{7}\right)\times\left(\left(\left(\dfrac{123}{5}\right)\sqrt{7}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{215}{8}\right)\sqrt{7}\right)+\left(14\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{63}{4}\right)\sqrt{7}\right)\left(\left(\dfrac{2619}{40}\right)\sqrt{7}+\dfrac{44}{7}\right)\\
&=&\left(\dfrac{164997}{160}\right)\sqrt{49}+\left(99\right)\sqrt{7}\\
&=&\dfrac{1154979}{160}+\left(99\right)\sqrt{7}\\
\end{eqnarray*}