L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-\dfrac{33}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{74}{9}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{13}{9}\right)\sqrt{4}\right)-\dfrac{62}{9}-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}\right)\right)\) et \( Y=\left(\dfrac{29}{8}\right)\sqrt{50}\) . 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(\left(-\dfrac{33}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{74}{9}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{13}{9}\right)\sqrt{4}\right)-\dfrac{62}{9}-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}\right)\right)\right)+\left(\left(\dfrac{29}{8}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\left(-33\right)\sqrt{2}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{148}{9}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}\right)-\left(-\dfrac{26}{9}-\dfrac{62}{9}-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}\right)\right)\right)+\left(\left(\dfrac{145}{8}\right)\sqrt{2}\right)\\
&=&\left(\left(\left(-33\right)\sqrt{2}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{148}{9}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}\right)-\left(-\dfrac{26}{9}-\dfrac{62}{9}-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}\right)\right)+\left(\dfrac{145}{8}\right)\sqrt{2}\\
&=&\left(-\dfrac{1199}{72}\right)\sqrt{2}+\dfrac{88}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{33}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{74}{9}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{13}{9}\right)\sqrt{4}\right)-\dfrac{62}{9}-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}\right)\right)\right)-\left(\left(\dfrac{29}{8}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\left(-33\right)\sqrt{2}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{148}{9}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}\right)-\left(-\dfrac{26}{9}-\dfrac{62}{9}-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}\right)\right)\right)-\left(\left(\dfrac{145}{8}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{313}{9}\right)\sqrt{2}+\dfrac{88}{9}\right)-\left(\left(\dfrac{145}{8}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{313}{9}\right)\sqrt{2}+\dfrac{88}{9}+\left(-\dfrac{145}{8}\right)\sqrt{2}\\
&=&\left(-\dfrac{3809}{72}\right)\sqrt{2}+\dfrac{88}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{33}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{74}{9}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{13}{9}\right)\sqrt{4}\right)-\dfrac{62}{9}-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}\right)\right)\right)\times\left(\left(\dfrac{29}{8}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\left(-33\right)\sqrt{2}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{148}{9}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}\right)-\left(-\dfrac{26}{9}-\dfrac{62}{9}-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}\right)\right)\right)\times\left(\left(\dfrac{145}{8}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{313}{9}\right)\sqrt{2}+\dfrac{88}{9}\right)\left(\left(\dfrac{145}{8}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{45385}{72}\right)\sqrt{4}+\left(\dfrac{1595}{9}\right)\sqrt{2}\\
&=&-\dfrac{45385}{36}+\left(\dfrac{1595}{9}\right)\sqrt{2}\\
\end{eqnarray*}