L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{10}{9}\) et \( Y=\dfrac{47}{8}+\left(-\dfrac{29}{5}\right)\sqrt{4}+\left(3\right)\sqrt{8}+\left(\dfrac{57}{8}\right)\sqrt{50}+\left(-\dfrac{25}{8}\right)\sqrt{50}+0+\left(-\dfrac{21}{4}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{10}{9}\right)+\left(\dfrac{47}{8}+\left(-\dfrac{29}{5}\right)\sqrt{4}+\left(3\right)\sqrt{8}+\left(\dfrac{57}{8}\right)\sqrt{50}+\left(-\dfrac{25}{8}\right)\sqrt{50}+0+\left(-\dfrac{21}{4}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{10}{9}\right)+\left(\dfrac{47}{8}-\dfrac{58}{5}+\left(6\right)\sqrt{2}+\left(\dfrac{285}{8}\right)\sqrt{2}+\left(-\dfrac{125}{8}\right)\sqrt{2}+0+\left(-\dfrac{63}{4}\right)\sqrt{2}\right)\\
&=&\dfrac{10}{9}+\dfrac{47}{8}-\dfrac{58}{5}+\left(6\right)\sqrt{2}+\left(\dfrac{285}{8}\right)\sqrt{2}+\left(-\dfrac{125}{8}\right)\sqrt{2}+0+\left(-\dfrac{63}{4}\right)\sqrt{2}\\
&=&-\dfrac{1661}{360}+\left(\dfrac{41}{4}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{10}{9}\right)-\left(\dfrac{47}{8}+\left(-\dfrac{29}{5}\right)\sqrt{4}+\left(3\right)\sqrt{8}+\left(\dfrac{57}{8}\right)\sqrt{50}+\left(-\dfrac{25}{8}\right)\sqrt{50}+0+\left(-\dfrac{21}{4}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{10}{9}\right)-\left(\dfrac{47}{8}-\dfrac{58}{5}+\left(6\right)\sqrt{2}+\left(\dfrac{285}{8}\right)\sqrt{2}+\left(-\dfrac{125}{8}\right)\sqrt{2}+0+\left(-\dfrac{63}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{10}{9}\right)-\left(-\dfrac{229}{40}+\left(\dfrac{41}{4}\right)\sqrt{2}\right)\\
&=&\dfrac{10}{9}+\dfrac{229}{40}+\left(-\dfrac{41}{4}\right)\sqrt{2}\\
&=&\dfrac{2461}{360}+\left(-\dfrac{41}{4}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{10}{9}\right)\times\left(\dfrac{47}{8}+\left(-\dfrac{29}{5}\right)\sqrt{4}+\left(3\right)\sqrt{8}+\left(\dfrac{57}{8}\right)\sqrt{50}+\left(-\dfrac{25}{8}\right)\sqrt{50}+0+\left(-\dfrac{21}{4}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{10}{9}\right)\times\left(\dfrac{47}{8}-\dfrac{58}{5}+\left(6\right)\sqrt{2}+\left(\dfrac{285}{8}\right)\sqrt{2}+\left(-\dfrac{125}{8}\right)\sqrt{2}+0+\left(-\dfrac{63}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{10}{9}\right)\left(-\dfrac{229}{40}+\left(\dfrac{41}{4}\right)\sqrt{2}\right)\\
&=&-\dfrac{229}{36}+\left(\dfrac{205}{18}\right)\sqrt{2}\\
&=&-\dfrac{229}{36}+\left(\dfrac{205}{18}\right)\sqrt{2}\\
\end{eqnarray*}