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 \( X=\left(\left(-\dfrac{23}{6}\right)\sqrt{20}\right)-\left(\left(\left(-\dfrac{61}{8}\right)\sqrt{20}\right)-\dfrac{5}{3}\right)\) et \( Y=-\dfrac{71}{6}-\left(\left(-\dfrac{78}{7}\right)\sqrt{45}\right)-\dfrac{77}{2}-\left(\left(0\right)\sqrt{45}\right)+\dfrac{79}{9}-\left(\left(4\right)\sqrt{20}\right)\) . 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{23}{6}\right)\sqrt{20}\right)-\left(\left(\left(-\dfrac{61}{8}\right)\sqrt{20}\right)-\dfrac{5}{3}\right)\right)+\left(-\dfrac{71}{6}-\left(\left(-\dfrac{78}{7}\right)\sqrt{45}\right)-\dfrac{77}{2}-\left(\left(0\right)\sqrt{45}\right)+\dfrac{79}{9}-\left(\left(4\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(-\dfrac{23}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{61}{4}\right)\sqrt{5}\right)-\dfrac{5}{3}\right)\right)+\left(-\dfrac{71}{6}-\left(\left(-\dfrac{234}{7}\right)\sqrt{5}\right)-\dfrac{77}{2}-\left(\left(0\right)\sqrt{5}\right)+\dfrac{79}{9}-\left(\left(8\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{23}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{61}{4}\right)\sqrt{5}\right)-\dfrac{5}{3}\right)-\dfrac{71}{6}-\left(\left(-\dfrac{234}{7}\right)\sqrt{5}\right)-\dfrac{77}{2}-\left(\left(0\right)\sqrt{5}\right)+\dfrac{79}{9}-\left(\left(8\right)\sqrt{5}\right)\\
&=&\left(\dfrac{2773}{84}\right)\sqrt{5}-\dfrac{359}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{23}{6}\right)\sqrt{20}\right)-\left(\left(\left(-\dfrac{61}{8}\right)\sqrt{20}\right)-\dfrac{5}{3}\right)\right)-\left(-\dfrac{71}{6}-\left(\left(-\dfrac{78}{7}\right)\sqrt{45}\right)-\dfrac{77}{2}-\left(\left(0\right)\sqrt{45}\right)+\dfrac{79}{9}-\left(\left(4\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(-\dfrac{23}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{61}{4}\right)\sqrt{5}\right)-\dfrac{5}{3}\right)\right)-\left(-\dfrac{71}{6}-\left(\left(-\dfrac{234}{7}\right)\sqrt{5}\right)-\dfrac{77}{2}-\left(\left(0\right)\sqrt{5}\right)+\dfrac{79}{9}-\left(\left(8\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{91}{12}\right)\sqrt{5}+\dfrac{5}{3}\right)-\left(-\dfrac{374}{9}+\left(\dfrac{178}{7}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{91}{12}\right)\sqrt{5}+\dfrac{5}{3}+\dfrac{374}{9}+\left(-\dfrac{178}{7}\right)\sqrt{5}\\
&=&\left(-\dfrac{1499}{84}\right)\sqrt{5}+\dfrac{389}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{23}{6}\right)\sqrt{20}\right)-\left(\left(\left(-\dfrac{61}{8}\right)\sqrt{20}\right)-\dfrac{5}{3}\right)\right)\times\left(-\dfrac{71}{6}-\left(\left(-\dfrac{78}{7}\right)\sqrt{45}\right)-\dfrac{77}{2}-\left(\left(0\right)\sqrt{45}\right)+\dfrac{79}{9}-\left(\left(4\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(-\dfrac{23}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{61}{4}\right)\sqrt{5}\right)-\dfrac{5}{3}\right)\right)\times\left(-\dfrac{71}{6}-\left(\left(-\dfrac{234}{7}\right)\sqrt{5}\right)-\dfrac{77}{2}-\left(\left(0\right)\sqrt{5}\right)+\dfrac{79}{9}-\left(\left(8\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{91}{12}\right)\sqrt{5}+\dfrac{5}{3}\right)\left(-\dfrac{374}{9}+\left(\dfrac{178}{7}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{103099}{378}\right)\sqrt{5}+\left(\dfrac{1157}{6}\right)\sqrt{25}-\dfrac{1870}{27}\\
&=&\left(-\dfrac{103099}{378}\right)\sqrt{5}+\dfrac{48325}{54}\\
\end{eqnarray*}