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 \( X=\left(\left(\dfrac{67}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{19}{8}\right)\sqrt{49}+\left(\dfrac{73}{6}\right)\sqrt{28}\right)-\left(\left(\left(-\dfrac{15}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{20}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{63}\right)\right)\) et \( Y=\left(-\dfrac{24}{5}\right)\sqrt{49}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\dfrac{67}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{19}{8}\right)\sqrt{49}+\left(\dfrac{73}{6}\right)\sqrt{28}\right)-\left(\left(\left(-\dfrac{15}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{20}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{63}\right)\right)\right)+\left(\left(-\dfrac{24}{5}\right)\sqrt{49}\right)\\
&=&\left(\left(\left(\dfrac{67}{3}\right)\sqrt{7}\right)-\left(\dfrac{133}{8}+\left(\dfrac{73}{3}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{100}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{7}\right)\right)\right)+\left(-\dfrac{168}{5}\right)\\
&=&\left(\left(\dfrac{67}{3}\right)\sqrt{7}\right)-\left(\dfrac{133}{8}+\left(\dfrac{73}{3}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{100}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{7}\right)\right)-\dfrac{168}{5}\\
&=&\left(\dfrac{727}{12}\right)\sqrt{7}-\dfrac{2009}{40}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{67}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{19}{8}\right)\sqrt{49}+\left(\dfrac{73}{6}\right)\sqrt{28}\right)-\left(\left(\left(-\dfrac{15}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{20}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{63}\right)\right)\right)-\left(\left(-\dfrac{24}{5}\right)\sqrt{49}\right)\\
&=&\left(\left(\left(\dfrac{67}{3}\right)\sqrt{7}\right)-\left(\dfrac{133}{8}+\left(\dfrac{73}{3}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{100}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{7}\right)\right)\right)-\left(-\dfrac{168}{5}\right)\\
&=&\left(\left(\dfrac{727}{12}\right)\sqrt{7}-\dfrac{133}{8}\right)-\left(-\dfrac{168}{5}\right)\\
&=&\left(\dfrac{727}{12}\right)\sqrt{7}-\dfrac{133}{8}+\dfrac{168}{5}\\
&=&\left(\dfrac{727}{12}\right)\sqrt{7}+\dfrac{679}{40}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{67}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{19}{8}\right)\sqrt{49}+\left(\dfrac{73}{6}\right)\sqrt{28}\right)-\left(\left(\left(-\dfrac{15}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{20}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{63}\right)\right)\right)\times\left(\left(-\dfrac{24}{5}\right)\sqrt{49}\right)\\
&=&\left(\left(\left(\dfrac{67}{3}\right)\sqrt{7}\right)-\left(\dfrac{133}{8}+\left(\dfrac{73}{3}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{100}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{7}\right)\right)\right)\times\left(-\dfrac{168}{5}\right)\\
&=&\left(\left(\dfrac{727}{12}\right)\sqrt{7}-\dfrac{133}{8}\right)\left(-\dfrac{168}{5}\right)\\
&=&\left(-\dfrac{10178}{5}\right)\sqrt{7}+\dfrac{2793}{5}\\
&=&\left(-\dfrac{10178}{5}\right)\sqrt{7}+\dfrac{2793}{5}\\
\end{eqnarray*}