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 \( X=-9+\left(\dfrac{9}{4}\right)\sqrt{8}+\left(5\right)\sqrt{4}+\left(\dfrac{25}{3}\right)\sqrt{8}+\left(0\right)\sqrt{8}\) et \( Y=\left(\left(\left(\dfrac{29}{5}\right)\sqrt{4}\right)-\dfrac{37}{4}\right)-\left(\left(\left(\dfrac{29}{5}\right)\sqrt{4}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{50}\right)\right)+2-\dfrac{14}{3}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-9+\left(\dfrac{9}{4}\right)\sqrt{8}+\left(5\right)\sqrt{4}+\left(\dfrac{25}{3}\right)\sqrt{8}+\left(0\right)\sqrt{8}\right)+\left(\left(\left(\left(\dfrac{29}{5}\right)\sqrt{4}\right)-\dfrac{37}{4}\right)-\left(\left(\left(\dfrac{29}{5}\right)\sqrt{4}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{50}\right)\right)+2-\dfrac{14}{3}\right)\\
&=&\left(-9+\left(\dfrac{9}{2}\right)\sqrt{2}+10+\left(\dfrac{50}{3}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)+\left(\left(\dfrac{58}{5}-\dfrac{37}{4}\right)-\left(\dfrac{58}{5}-\left(\left(\dfrac{145}{4}\right)\sqrt{2}\right)\right)+2-\dfrac{14}{3}\right)\\
&=&-9+\left(\dfrac{9}{2}\right)\sqrt{2}+10+\left(\dfrac{50}{3}\right)\sqrt{2}+\left(0\right)\sqrt{2}+\left(\dfrac{58}{5}-\dfrac{37}{4}\right)-\left(\dfrac{58}{5}-\left(\left(\dfrac{145}{4}\right)\sqrt{2}\right)\right)+2-\dfrac{14}{3}\\
&=&-\dfrac{131}{12}+\left(\dfrac{689}{12}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-9+\left(\dfrac{9}{4}\right)\sqrt{8}+\left(5\right)\sqrt{4}+\left(\dfrac{25}{3}\right)\sqrt{8}+\left(0\right)\sqrt{8}\right)-\left(\left(\left(\left(\dfrac{29}{5}\right)\sqrt{4}\right)-\dfrac{37}{4}\right)-\left(\left(\left(\dfrac{29}{5}\right)\sqrt{4}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{50}\right)\right)+2-\dfrac{14}{3}\right)\\
&=&\left(-9+\left(\dfrac{9}{2}\right)\sqrt{2}+10+\left(\dfrac{50}{3}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{58}{5}-\dfrac{37}{4}\right)-\left(\dfrac{58}{5}-\left(\left(\dfrac{145}{4}\right)\sqrt{2}\right)\right)+2-\dfrac{14}{3}\right)\\
&=&\left(1+\left(\dfrac{127}{6}\right)\sqrt{2}\right)-\left(-\dfrac{143}{12}+\left(\dfrac{145}{4}\right)\sqrt{2}\right)\\
&=&1+\left(\dfrac{127}{6}\right)\sqrt{2}+\dfrac{143}{12}+\left(-\dfrac{145}{4}\right)\sqrt{2}\\
&=&\dfrac{155}{12}+\left(-\dfrac{181}{12}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-9+\left(\dfrac{9}{4}\right)\sqrt{8}+\left(5\right)\sqrt{4}+\left(\dfrac{25}{3}\right)\sqrt{8}+\left(0\right)\sqrt{8}\right)\times\left(\left(\left(\left(\dfrac{29}{5}\right)\sqrt{4}\right)-\dfrac{37}{4}\right)-\left(\left(\left(\dfrac{29}{5}\right)\sqrt{4}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{50}\right)\right)+2-\dfrac{14}{3}\right)\\
&=&\left(-9+\left(\dfrac{9}{2}\right)\sqrt{2}+10+\left(\dfrac{50}{3}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\times\left(\left(\dfrac{58}{5}-\dfrac{37}{4}\right)-\left(\dfrac{58}{5}-\left(\left(\dfrac{145}{4}\right)\sqrt{2}\right)\right)+2-\dfrac{14}{3}\right)\\
&=&\left(1+\left(\dfrac{127}{6}\right)\sqrt{2}\right)\left(-\dfrac{143}{12}+\left(\dfrac{145}{4}\right)\sqrt{2}\right)\\
&=&-\dfrac{143}{12}+\left(-\dfrac{15551}{72}\right)\sqrt{2}+\left(\dfrac{18415}{24}\right)\sqrt{4}\\
&=&\dfrac{4568}{3}+\left(-\dfrac{15551}{72}\right)\sqrt{2}\\
\end{eqnarray*}