L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{23}{8}\right)\sqrt{18}\right)+\dfrac{34}{9}-\left(\dfrac{65}{4}-\left(\left(-\dfrac{27}{4}\right)\sqrt{4}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{50}\right)\right)\) et \( Y=\left(\dfrac{43}{4}\right)\sqrt{4}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{23}{8}\right)\sqrt{18}\right)+\dfrac{34}{9}-\left(\dfrac{65}{4}-\left(\left(-\dfrac{27}{4}\right)\sqrt{4}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{50}\right)\right)\right)+\left(\left(\dfrac{43}{4}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-\dfrac{69}{8}\right)\sqrt{2}\right)+\dfrac{34}{9}-\left(\dfrac{65}{4}+\dfrac{27}{2}-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{2}\right)\right)\right)+\left(\dfrac{43}{2}\right)\\
&=&\left(\left(-\dfrac{69}{8}\right)\sqrt{2}\right)+\dfrac{34}{9}-\left(\dfrac{65}{4}+\dfrac{27}{2}-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{2}\right)\right)+\dfrac{43}{2}\\
&=&\left(-\dfrac{647}{24}\right)\sqrt{2}-\dfrac{161}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{23}{8}\right)\sqrt{18}\right)+\dfrac{34}{9}-\left(\dfrac{65}{4}-\left(\left(-\dfrac{27}{4}\right)\sqrt{4}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{50}\right)\right)\right)-\left(\left(\dfrac{43}{4}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-\dfrac{69}{8}\right)\sqrt{2}\right)+\dfrac{34}{9}-\left(\dfrac{65}{4}+\dfrac{27}{2}-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{2}\right)\right)\right)-\left(\dfrac{43}{2}\right)\\
&=&\left(\left(-\dfrac{647}{24}\right)\sqrt{2}-\dfrac{935}{36}\right)-\left(\dfrac{43}{2}\right)\\
&=&\left(-\dfrac{647}{24}\right)\sqrt{2}-\dfrac{935}{36}+-\dfrac{43}{2}\\
&=&\left(-\dfrac{647}{24}\right)\sqrt{2}-\dfrac{1709}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{23}{8}\right)\sqrt{18}\right)+\dfrac{34}{9}-\left(\dfrac{65}{4}-\left(\left(-\dfrac{27}{4}\right)\sqrt{4}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{50}\right)\right)\right)\times\left(\left(\dfrac{43}{4}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-\dfrac{69}{8}\right)\sqrt{2}\right)+\dfrac{34}{9}-\left(\dfrac{65}{4}+\dfrac{27}{2}-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{2}\right)\right)\right)\times\left(\dfrac{43}{2}\right)\\
&=&\left(\left(-\dfrac{647}{24}\right)\sqrt{2}-\dfrac{935}{36}\right)\left(\dfrac{43}{2}\right)\\
&=&\left(-\dfrac{27821}{48}\right)\sqrt{2}-\dfrac{40205}{72}\\
&=&\left(-\dfrac{27821}{48}\right)\sqrt{2}-\dfrac{40205}{72}\\
\end{eqnarray*}