L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{73}{9}-\left(\left(\dfrac{38}{5}\right)\sqrt{4}\right)\right)+\dfrac{61}{4}-\left(\left(-3\right)\sqrt{8}+\left(\dfrac{19}{3}\right)\sqrt{8}\right)\) et \( Y=\left(-\dfrac{18}{7}\right)\sqrt{4}+\left(\dfrac{29}{4}\right)\sqrt{18}+\left(-7\right)\sqrt{4}\) . 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}{9}-\left(\left(\dfrac{38}{5}\right)\sqrt{4}\right)\right)+\dfrac{61}{4}-\left(\left(-3\right)\sqrt{8}+\left(\dfrac{19}{3}\right)\sqrt{8}\right)\right)+\left(\left(-\dfrac{18}{7}\right)\sqrt{4}+\left(\dfrac{29}{4}\right)\sqrt{18}+\left(-7\right)\sqrt{4}\right)\\
&=&\left(\left(-\dfrac{73}{9}-\dfrac{76}{5}\right)+\dfrac{61}{4}-\left(\left(-6\right)\sqrt{2}+\left(\dfrac{38}{3}\right)\sqrt{2}\right)\right)+\left(-\dfrac{36}{7}+\left(\dfrac{87}{4}\right)\sqrt{2}-14\right)\\
&=&\left(-\dfrac{73}{9}-\dfrac{76}{5}\right)+\dfrac{61}{4}-\left(\left(-6\right)\sqrt{2}+\left(\dfrac{38}{3}\right)\sqrt{2}\right)-\dfrac{36}{7}+\left(\dfrac{87}{4}\right)\sqrt{2}-14\\
&=&-\dfrac{34277}{1260}+\left(\dfrac{181}{12}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{73}{9}-\left(\left(\dfrac{38}{5}\right)\sqrt{4}\right)\right)+\dfrac{61}{4}-\left(\left(-3\right)\sqrt{8}+\left(\dfrac{19}{3}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{4}+\left(\dfrac{29}{4}\right)\sqrt{18}+\left(-7\right)\sqrt{4}\right)\\
&=&\left(\left(-\dfrac{73}{9}-\dfrac{76}{5}\right)+\dfrac{61}{4}-\left(\left(-6\right)\sqrt{2}+\left(\dfrac{38}{3}\right)\sqrt{2}\right)\right)-\left(-\dfrac{36}{7}+\left(\dfrac{87}{4}\right)\sqrt{2}-14\right)\\
&=&\left(-\dfrac{1451}{180}+\left(-\dfrac{20}{3}\right)\sqrt{2}\right)-\left(-\dfrac{134}{7}+\left(\dfrac{87}{4}\right)\sqrt{2}\right)\\
&=&-\dfrac{1451}{180}+\left(-\dfrac{20}{3}\right)\sqrt{2}+\dfrac{134}{7}+\left(-\dfrac{87}{4}\right)\sqrt{2}\\
&=&\dfrac{13963}{1260}+\left(-\dfrac{341}{12}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{73}{9}-\left(\left(\dfrac{38}{5}\right)\sqrt{4}\right)\right)+\dfrac{61}{4}-\left(\left(-3\right)\sqrt{8}+\left(\dfrac{19}{3}\right)\sqrt{8}\right)\right)\times\left(\left(-\dfrac{18}{7}\right)\sqrt{4}+\left(\dfrac{29}{4}\right)\sqrt{18}+\left(-7\right)\sqrt{4}\right)\\
&=&\left(\left(-\dfrac{73}{9}-\dfrac{76}{5}\right)+\dfrac{61}{4}-\left(\left(-6\right)\sqrt{2}+\left(\dfrac{38}{3}\right)\sqrt{2}\right)\right)\times\left(-\dfrac{36}{7}+\left(\dfrac{87}{4}\right)\sqrt{2}-14\right)\\
&=&\left(-\dfrac{1451}{180}+\left(-\dfrac{20}{3}\right)\sqrt{2}\right)\left(-\dfrac{134}{7}+\left(\dfrac{87}{4}\right)\sqrt{2}\right)\\
&=&\dfrac{97217}{630}+\left(-\dfrac{80153}{1680}\right)\sqrt{2}+\left(-145\right)\sqrt{4}\\
&=&-\dfrac{85483}{630}+\left(-\dfrac{80153}{1680}\right)\sqrt{2}\\
\end{eqnarray*}