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 \( X=\left(\left(5\right)\sqrt{27}\right)-\left(\left(\dfrac{67}{5}\right)\sqrt{27}\right)+\dfrac{25}{3}+\left(-\dfrac{17}{4}\right)\sqrt{27}\) et \( Y=\left(-\dfrac{8}{7}\right)\sqrt{12}\) . 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(5\right)\sqrt{27}\right)-\left(\left(\dfrac{67}{5}\right)\sqrt{27}\right)+\dfrac{25}{3}+\left(-\dfrac{17}{4}\right)\sqrt{27}\right)+\left(\left(-\dfrac{8}{7}\right)\sqrt{12}\right)\\
&=&\left(\left(\left(15\right)\sqrt{3}\right)-\left(\left(\dfrac{201}{5}\right)\sqrt{3}\right)+\dfrac{25}{3}+\left(-\dfrac{51}{4}\right)\sqrt{3}\right)+\left(\left(-\dfrac{16}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(15\right)\sqrt{3}\right)-\left(\left(\dfrac{201}{5}\right)\sqrt{3}\right)+\dfrac{25}{3}+\left(-\dfrac{51}{4}\right)\sqrt{3}+\left(-\dfrac{16}{7}\right)\sqrt{3}\\
&=&\left(-\dfrac{5633}{140}\right)\sqrt{3}+\dfrac{25}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(5\right)\sqrt{27}\right)-\left(\left(\dfrac{67}{5}\right)\sqrt{27}\right)+\dfrac{25}{3}+\left(-\dfrac{17}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{8}{7}\right)\sqrt{12}\right)\\
&=&\left(\left(\left(15\right)\sqrt{3}\right)-\left(\left(\dfrac{201}{5}\right)\sqrt{3}\right)+\dfrac{25}{3}+\left(-\dfrac{51}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{16}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{759}{20}\right)\sqrt{3}+\dfrac{25}{3}\right)-\left(\left(-\dfrac{16}{7}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{759}{20}\right)\sqrt{3}+\dfrac{25}{3}+\left(\dfrac{16}{7}\right)\sqrt{3}\\
&=&\left(-\dfrac{4993}{140}\right)\sqrt{3}+\dfrac{25}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(5\right)\sqrt{27}\right)-\left(\left(\dfrac{67}{5}\right)\sqrt{27}\right)+\dfrac{25}{3}+\left(-\dfrac{17}{4}\right)\sqrt{27}\right)\times\left(\left(-\dfrac{8}{7}\right)\sqrt{12}\right)\\
&=&\left(\left(\left(15\right)\sqrt{3}\right)-\left(\left(\dfrac{201}{5}\right)\sqrt{3}\right)+\dfrac{25}{3}+\left(-\dfrac{51}{4}\right)\sqrt{3}\right)\times\left(\left(-\dfrac{16}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{759}{20}\right)\sqrt{3}+\dfrac{25}{3}\right)\left(\left(-\dfrac{16}{7}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{3036}{35}\right)\sqrt{9}+\left(-\dfrac{400}{21}\right)\sqrt{3}\\
&=&\dfrac{9108}{35}+\left(-\dfrac{400}{21}\right)\sqrt{3}\\
\end{eqnarray*}