L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{44}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{50}\right)+\left(-\dfrac{71}{9}\right)\sqrt{50}+\dfrac{32}{7}+\left(6\right)\sqrt{8}+\left(\dfrac{31}{3}\right)\sqrt{4}\) et \( Y=\left(\dfrac{46}{7}\right)\sqrt{4}\) . 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{44}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{50}\right)+\left(-\dfrac{71}{9}\right)\sqrt{50}+\dfrac{32}{7}+\left(6\right)\sqrt{8}+\left(\dfrac{31}{3}\right)\sqrt{4}\right)+\left(\left(\dfrac{46}{7}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-\dfrac{220}{7}\right)\sqrt{2}\right)-\left(\left(-28\right)\sqrt{2}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{2}\right)+\left(-\dfrac{355}{9}\right)\sqrt{2}+\dfrac{32}{7}+\left(12\right)\sqrt{2}+\dfrac{62}{3}\right)+\left(\dfrac{92}{7}\right)\\
&=&\left(\left(-\dfrac{220}{7}\right)\sqrt{2}\right)-\left(\left(-28\right)\sqrt{2}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{2}\right)+\left(-\dfrac{355}{9}\right)\sqrt{2}+\dfrac{32}{7}+\left(12\right)\sqrt{2}+\dfrac{62}{3}+\dfrac{92}{7}\\
&=&\left(-\dfrac{2785}{63}\right)\sqrt{2}+\dfrac{806}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{44}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{50}\right)+\left(-\dfrac{71}{9}\right)\sqrt{50}+\dfrac{32}{7}+\left(6\right)\sqrt{8}+\left(\dfrac{31}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{46}{7}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-\dfrac{220}{7}\right)\sqrt{2}\right)-\left(\left(-28\right)\sqrt{2}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{2}\right)+\left(-\dfrac{355}{9}\right)\sqrt{2}+\dfrac{32}{7}+\left(12\right)\sqrt{2}+\dfrac{62}{3}\right)-\left(\dfrac{92}{7}\right)\\
&=&\left(\left(-\dfrac{2785}{63}\right)\sqrt{2}+\dfrac{530}{21}\right)-\left(\dfrac{92}{7}\right)\\
&=&\left(-\dfrac{2785}{63}\right)\sqrt{2}+\dfrac{530}{21}+-\dfrac{92}{7}\\
&=&\left(-\dfrac{2785}{63}\right)\sqrt{2}+\dfrac{254}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{44}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{50}\right)+\left(-\dfrac{71}{9}\right)\sqrt{50}+\dfrac{32}{7}+\left(6\right)\sqrt{8}+\left(\dfrac{31}{3}\right)\sqrt{4}\right)\times\left(\left(\dfrac{46}{7}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-\dfrac{220}{7}\right)\sqrt{2}\right)-\left(\left(-28\right)\sqrt{2}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{2}\right)+\left(-\dfrac{355}{9}\right)\sqrt{2}+\dfrac{32}{7}+\left(12\right)\sqrt{2}+\dfrac{62}{3}\right)\times\left(\dfrac{92}{7}\right)\\
&=&\left(\left(-\dfrac{2785}{63}\right)\sqrt{2}+\dfrac{530}{21}\right)\left(\dfrac{92}{7}\right)\\
&=&\left(-\dfrac{256220}{441}\right)\sqrt{2}+\dfrac{48760}{147}\\
&=&\left(-\dfrac{256220}{441}\right)\sqrt{2}+\dfrac{48760}{147}\\
\end{eqnarray*}