L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{65}{9}\right)\sqrt{28}\) et \( Y=\left(\dfrac{11}{3}\right)\sqrt{49}+\left(-\dfrac{72}{5}\right)\sqrt{28}+\left(-\dfrac{71}{8}\right)\sqrt{175}+\left(-\dfrac{40}{9}\right)\sqrt{49}\) . 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{65}{9}\right)\sqrt{28}\right)+\left(\left(\dfrac{11}{3}\right)\sqrt{49}+\left(-\dfrac{72}{5}\right)\sqrt{28}+\left(-\dfrac{71}{8}\right)\sqrt{175}+\left(-\dfrac{40}{9}\right)\sqrt{49}\right)\\
&=&\left(\left(\dfrac{130}{9}\right)\sqrt{7}\right)+\left(\dfrac{77}{3}+\left(-\dfrac{144}{5}\right)\sqrt{7}+\left(-\dfrac{355}{8}\right)\sqrt{7}-\dfrac{280}{9}\right)\\
&=&\left(\dfrac{130}{9}\right)\sqrt{7}+\dfrac{77}{3}+\left(-\dfrac{144}{5}\right)\sqrt{7}+\left(-\dfrac{355}{8}\right)\sqrt{7}-\dfrac{280}{9}\\
&=&\left(-\dfrac{21143}{360}\right)\sqrt{7}-\dfrac{49}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{65}{9}\right)\sqrt{28}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{49}+\left(-\dfrac{72}{5}\right)\sqrt{28}+\left(-\dfrac{71}{8}\right)\sqrt{175}+\left(-\dfrac{40}{9}\right)\sqrt{49}\right)\\
&=&\left(\left(\dfrac{130}{9}\right)\sqrt{7}\right)-\left(\dfrac{77}{3}+\left(-\dfrac{144}{5}\right)\sqrt{7}+\left(-\dfrac{355}{8}\right)\sqrt{7}-\dfrac{280}{9}\right)\\
&=&\left(\left(\dfrac{130}{9}\right)\sqrt{7}\right)-\left(-\dfrac{49}{9}+\left(-\dfrac{2927}{40}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{130}{9}\right)\sqrt{7}+\dfrac{49}{9}+\left(\dfrac{2927}{40}\right)\sqrt{7}\\
&=&\left(\dfrac{31543}{360}\right)\sqrt{7}+\dfrac{49}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{65}{9}\right)\sqrt{28}\right)\times\left(\left(\dfrac{11}{3}\right)\sqrt{49}+\left(-\dfrac{72}{5}\right)\sqrt{28}+\left(-\dfrac{71}{8}\right)\sqrt{175}+\left(-\dfrac{40}{9}\right)\sqrt{49}\right)\\
&=&\left(\left(\dfrac{130}{9}\right)\sqrt{7}\right)\times\left(\dfrac{77}{3}+\left(-\dfrac{144}{5}\right)\sqrt{7}+\left(-\dfrac{355}{8}\right)\sqrt{7}-\dfrac{280}{9}\right)\\
&=&\left(\left(\dfrac{130}{9}\right)\sqrt{7}\right)\left(-\dfrac{49}{9}+\left(-\dfrac{2927}{40}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{6370}{81}\right)\sqrt{7}+\left(-\dfrac{38051}{36}\right)\sqrt{49}\\
&=&\left(-\dfrac{6370}{81}\right)\sqrt{7}-\dfrac{266357}{36}\\
\end{eqnarray*}