L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{39}{2}+\left(\dfrac{54}{7}\right)\sqrt{18}-\dfrac{5}{3}+1\right)-\left(-\dfrac{63}{8}-\left(\left(-\dfrac{39}{4}\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{8}\right)\right)-\left(\left(\left(\dfrac{22}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{50}\right)\right)\) et \( Y=\left(\dfrac{25}{9}\right)\sqrt{18}\) . 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{39}{2}+\left(\dfrac{54}{7}\right)\sqrt{18}-\dfrac{5}{3}+1\right)-\left(-\dfrac{63}{8}-\left(\left(-\dfrac{39}{4}\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{8}\right)\right)-\left(\left(\left(\dfrac{22}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{50}\right)\right)\right)+\left(\left(\dfrac{25}{9}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{39}{2}+\left(\dfrac{162}{7}\right)\sqrt{2}-\dfrac{5}{3}+1\right)-\left(-\dfrac{63}{8}+\dfrac{39}{2}-\left(\left(16\right)\sqrt{2}\right)\right)-\left(\left(\left(\dfrac{44}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{2}\right)\right)\right)+\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{39}{2}+\left(\dfrac{162}{7}\right)\sqrt{2}-\dfrac{5}{3}+1\right)-\left(-\dfrac{63}{8}+\dfrac{39}{2}-\left(\left(16\right)\sqrt{2}\right)\right)-\left(\left(\left(\dfrac{44}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{2}\right)\right)+\left(\dfrac{25}{3}\right)\sqrt{2}\\
&=&\dfrac{173}{24}+\left(\dfrac{1144}{21}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{39}{2}+\left(\dfrac{54}{7}\right)\sqrt{18}-\dfrac{5}{3}+1\right)-\left(-\dfrac{63}{8}-\left(\left(-\dfrac{39}{4}\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{8}\right)\right)-\left(\left(\left(\dfrac{22}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{50}\right)\right)\right)-\left(\left(\dfrac{25}{9}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{39}{2}+\left(\dfrac{162}{7}\right)\sqrt{2}-\dfrac{5}{3}+1\right)-\left(-\dfrac{63}{8}+\dfrac{39}{2}-\left(\left(16\right)\sqrt{2}\right)\right)-\left(\left(\left(\dfrac{44}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{2}\right)\right)\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{173}{24}+\left(\dfrac{323}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)\\
&=&\dfrac{173}{24}+\left(\dfrac{323}{7}\right)\sqrt{2}+\left(-\dfrac{25}{3}\right)\sqrt{2}\\
&=&\dfrac{173}{24}+\left(\dfrac{794}{21}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{39}{2}+\left(\dfrac{54}{7}\right)\sqrt{18}-\dfrac{5}{3}+1\right)-\left(-\dfrac{63}{8}-\left(\left(-\dfrac{39}{4}\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{8}\right)\right)-\left(\left(\left(\dfrac{22}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{50}\right)\right)\right)\times\left(\left(\dfrac{25}{9}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{39}{2}+\left(\dfrac{162}{7}\right)\sqrt{2}-\dfrac{5}{3}+1\right)-\left(-\dfrac{63}{8}+\dfrac{39}{2}-\left(\left(16\right)\sqrt{2}\right)\right)-\left(\left(\left(\dfrac{44}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{2}\right)\right)\right)\times\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{173}{24}+\left(\dfrac{323}{7}\right)\sqrt{2}\right)\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{4325}{72}\right)\sqrt{2}+\left(\dfrac{8075}{21}\right)\sqrt{4}\\
&=&\left(\dfrac{4325}{72}\right)\sqrt{2}+\dfrac{16150}{21}\\
\end{eqnarray*}