L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{43}{7}\right)\sqrt{8}+\left(\dfrac{29}{4}\right)\sqrt{4}+\left(-\dfrac{64}{7}\right)\sqrt{8}+\left(\dfrac{29}{4}\right)\sqrt{4}+\left(\dfrac{41}{2}\right)\sqrt{4}+\left(\dfrac{26}{9}\right)\sqrt{50}+\left(\left(\dfrac{63}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{41}{2}\right)\sqrt{4}\right)+\left(\dfrac{67}{2}\right)\sqrt{8}+\left(0\right)\sqrt{18}+\dfrac{35}{3}+\left(-9\right)\sqrt{4}-\dfrac{31}{2}\) et \( Y=\left(\dfrac{7}{4}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{43}{7}\right)\sqrt{8}+\left(\dfrac{29}{4}\right)\sqrt{4}+\left(-\dfrac{64}{7}\right)\sqrt{8}+\left(\dfrac{29}{4}\right)\sqrt{4}+\left(\dfrac{41}{2}\right)\sqrt{4}+\left(\dfrac{26}{9}\right)\sqrt{50}+\left(\left(\dfrac{63}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{41}{2}\right)\sqrt{4}\right)+\left(\dfrac{67}{2}\right)\sqrt{8}+\left(0\right)\sqrt{18}+\dfrac{35}{3}+\left(-9\right)\sqrt{4}-\dfrac{31}{2}\right)+\left(\left(\dfrac{7}{4}\right)\sqrt{18}\right)\\
&=&\left(\left(-\dfrac{86}{7}\right)\sqrt{2}+\dfrac{29}{2}+\left(-\dfrac{128}{7}\right)\sqrt{2}+\dfrac{29}{2}+41+\left(\dfrac{130}{9}\right)\sqrt{2}+63-41+\left(67\right)\sqrt{2}+\left(0\right)\sqrt{2}+\dfrac{35}{3}-18-\dfrac{31}{2}\right)+\left(\left(\dfrac{21}{4}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{86}{7}\right)\sqrt{2}+\dfrac{29}{2}+\left(-\dfrac{128}{7}\right)\sqrt{2}+\dfrac{29}{2}+41+\left(\dfrac{130}{9}\right)\sqrt{2}+63-41+\left(67\right)\sqrt{2}+\left(0\right)\sqrt{2}+\dfrac{35}{3}-18-\dfrac{31}{2}+\left(\dfrac{21}{4}\right)\sqrt{2}\\
&=&\left(\dfrac{14143}{252}\right)\sqrt{2}+\dfrac{421}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{43}{7}\right)\sqrt{8}+\left(\dfrac{29}{4}\right)\sqrt{4}+\left(-\dfrac{64}{7}\right)\sqrt{8}+\left(\dfrac{29}{4}\right)\sqrt{4}+\left(\dfrac{41}{2}\right)\sqrt{4}+\left(\dfrac{26}{9}\right)\sqrt{50}+\left(\left(\dfrac{63}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{41}{2}\right)\sqrt{4}\right)+\left(\dfrac{67}{2}\right)\sqrt{8}+\left(0\right)\sqrt{18}+\dfrac{35}{3}+\left(-9\right)\sqrt{4}-\dfrac{31}{2}\right)-\left(\left(\dfrac{7}{4}\right)\sqrt{18}\right)\\
&=&\left(\left(-\dfrac{86}{7}\right)\sqrt{2}+\dfrac{29}{2}+\left(-\dfrac{128}{7}\right)\sqrt{2}+\dfrac{29}{2}+41+\left(\dfrac{130}{9}\right)\sqrt{2}+63-41+\left(67\right)\sqrt{2}+\left(0\right)\sqrt{2}+\dfrac{35}{3}-18-\dfrac{31}{2}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{3205}{63}\right)\sqrt{2}+\dfrac{421}{6}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{3205}{63}\right)\sqrt{2}+\dfrac{421}{6}+\left(-\dfrac{21}{4}\right)\sqrt{2}\\
&=&\left(\dfrac{11497}{252}\right)\sqrt{2}+\dfrac{421}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{43}{7}\right)\sqrt{8}+\left(\dfrac{29}{4}\right)\sqrt{4}+\left(-\dfrac{64}{7}\right)\sqrt{8}+\left(\dfrac{29}{4}\right)\sqrt{4}+\left(\dfrac{41}{2}\right)\sqrt{4}+\left(\dfrac{26}{9}\right)\sqrt{50}+\left(\left(\dfrac{63}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{41}{2}\right)\sqrt{4}\right)+\left(\dfrac{67}{2}\right)\sqrt{8}+\left(0\right)\sqrt{18}+\dfrac{35}{3}+\left(-9\right)\sqrt{4}-\dfrac{31}{2}\right)\times\left(\left(\dfrac{7}{4}\right)\sqrt{18}\right)\\
&=&\left(\left(-\dfrac{86}{7}\right)\sqrt{2}+\dfrac{29}{2}+\left(-\dfrac{128}{7}\right)\sqrt{2}+\dfrac{29}{2}+41+\left(\dfrac{130}{9}\right)\sqrt{2}+63-41+\left(67\right)\sqrt{2}+\left(0\right)\sqrt{2}+\dfrac{35}{3}-18-\dfrac{31}{2}\right)\times\left(\left(\dfrac{21}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{3205}{63}\right)\sqrt{2}+\dfrac{421}{6}\right)\left(\left(\dfrac{21}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{3205}{12}\right)\sqrt{4}+\left(\dfrac{2947}{8}\right)\sqrt{2}\\
&=&\dfrac{3205}{6}+\left(\dfrac{2947}{8}\right)\sqrt{2}\\
\end{eqnarray*}