L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{73}{3}\right)\sqrt{8}\) et \( Y=\left(-9\right)\sqrt{8}+\left(-\dfrac{31}{5}\right)\sqrt{50}-\dfrac{13}{9}+\left(\dfrac{57}{7}\right)\sqrt{18}+\left(-9\right)\sqrt{50}+\left(9\right)\sqrt{18}+\left(\dfrac{77}{5}\right)\sqrt{18}+\left(\left(0\right)\sqrt{18}\right)+\dfrac{47}{7}\) . 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{73}{3}\right)\sqrt{8}\right)+\left(\left(-9\right)\sqrt{8}+\left(-\dfrac{31}{5}\right)\sqrt{50}-\dfrac{13}{9}+\left(\dfrac{57}{7}\right)\sqrt{18}+\left(-9\right)\sqrt{50}+\left(9\right)\sqrt{18}+\left(\dfrac{77}{5}\right)\sqrt{18}+\left(\left(0\right)\sqrt{18}\right)+\dfrac{47}{7}\right)\\
&=&\left(\left(\dfrac{146}{3}\right)\sqrt{2}\right)+\left(\left(-18\right)\sqrt{2}+\left(-31\right)\sqrt{2}-\dfrac{13}{9}+\left(\dfrac{171}{7}\right)\sqrt{2}+\left(-45\right)\sqrt{2}+\left(27\right)\sqrt{2}+\left(\dfrac{231}{5}\right)\sqrt{2}+\left(\left(0\right)\sqrt{2}\right)+\dfrac{47}{7}\right)\\
&=&\left(\dfrac{146}{3}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\left(-31\right)\sqrt{2}-\dfrac{13}{9}+\left(\dfrac{171}{7}\right)\sqrt{2}+\left(-45\right)\sqrt{2}+\left(27\right)\sqrt{2}+\left(\dfrac{231}{5}\right)\sqrt{2}+\left(\left(0\right)\sqrt{2}\right)+\dfrac{47}{7}\\
&=&\left(\dfrac{5491}{105}\right)\sqrt{2}+\dfrac{332}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{73}{3}\right)\sqrt{8}\right)-\left(\left(-9\right)\sqrt{8}+\left(-\dfrac{31}{5}\right)\sqrt{50}-\dfrac{13}{9}+\left(\dfrac{57}{7}\right)\sqrt{18}+\left(-9\right)\sqrt{50}+\left(9\right)\sqrt{18}+\left(\dfrac{77}{5}\right)\sqrt{18}+\left(\left(0\right)\sqrt{18}\right)+\dfrac{47}{7}\right)\\
&=&\left(\left(\dfrac{146}{3}\right)\sqrt{2}\right)-\left(\left(-18\right)\sqrt{2}+\left(-31\right)\sqrt{2}-\dfrac{13}{9}+\left(\dfrac{171}{7}\right)\sqrt{2}+\left(-45\right)\sqrt{2}+\left(27\right)\sqrt{2}+\left(\dfrac{231}{5}\right)\sqrt{2}+\left(\left(0\right)\sqrt{2}\right)+\dfrac{47}{7}\right)\\
&=&\left(\left(\dfrac{146}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{127}{35}\right)\sqrt{2}+\dfrac{332}{63}\right)\\
&=&\left(\dfrac{146}{3}\right)\sqrt{2}+\left(-\dfrac{127}{35}\right)\sqrt{2}-\dfrac{332}{63}\\
&=&\left(\dfrac{4729}{105}\right)\sqrt{2}-\dfrac{332}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{73}{3}\right)\sqrt{8}\right)\times\left(\left(-9\right)\sqrt{8}+\left(-\dfrac{31}{5}\right)\sqrt{50}-\dfrac{13}{9}+\left(\dfrac{57}{7}\right)\sqrt{18}+\left(-9\right)\sqrt{50}+\left(9\right)\sqrt{18}+\left(\dfrac{77}{5}\right)\sqrt{18}+\left(\left(0\right)\sqrt{18}\right)+\dfrac{47}{7}\right)\\
&=&\left(\left(\dfrac{146}{3}\right)\sqrt{2}\right)\times\left(\left(-18\right)\sqrt{2}+\left(-31\right)\sqrt{2}-\dfrac{13}{9}+\left(\dfrac{171}{7}\right)\sqrt{2}+\left(-45\right)\sqrt{2}+\left(27\right)\sqrt{2}+\left(\dfrac{231}{5}\right)\sqrt{2}+\left(\left(0\right)\sqrt{2}\right)+\dfrac{47}{7}\right)\\
&=&\left(\left(\dfrac{146}{3}\right)\sqrt{2}\right)\left(\left(\dfrac{127}{35}\right)\sqrt{2}+\dfrac{332}{63}\right)\\
&=&\left(\dfrac{18542}{105}\right)\sqrt{4}+\left(\dfrac{48472}{189}\right)\sqrt{2}\\
&=&\dfrac{37084}{105}+\left(\dfrac{48472}{189}\right)\sqrt{2}\\
\end{eqnarray*}