L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{29}{3}\right)\sqrt{4}+\left(-9\right)\sqrt{50}+\left(\dfrac{67}{7}\right)\sqrt{18}+\left(-\dfrac{71}{7}\right)\sqrt{8}+\left(-\dfrac{23}{4}\right)\sqrt{4}\) et \( Y=\left(-\dfrac{76}{7}\right)\sqrt{50}+0-\left(\left(-\dfrac{22}{7}\right)\sqrt{50}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{29}{3}\right)\sqrt{4}+\left(-9\right)\sqrt{50}+\left(\dfrac{67}{7}\right)\sqrt{18}+\left(-\dfrac{71}{7}\right)\sqrt{8}+\left(-\dfrac{23}{4}\right)\sqrt{4}\right)+\left(\left(-\dfrac{76}{7}\right)\sqrt{50}+0-\left(\left(-\dfrac{22}{7}\right)\sqrt{50}\right)\right)\\
&=&\left(\dfrac{58}{3}+\left(-45\right)\sqrt{2}+\left(\dfrac{201}{7}\right)\sqrt{2}+\left(-\dfrac{142}{7}\right)\sqrt{2}-\dfrac{23}{2}\right)+\left(\left(-\dfrac{380}{7}\right)\sqrt{2}+0-\left(\left(-\dfrac{110}{7}\right)\sqrt{2}\right)\right)\\
&=&\dfrac{58}{3}+\left(-45\right)\sqrt{2}+\left(\dfrac{201}{7}\right)\sqrt{2}+\left(-\dfrac{142}{7}\right)\sqrt{2}-\dfrac{23}{2}+\left(-\dfrac{380}{7}\right)\sqrt{2}+0-\left(\left(-\dfrac{110}{7}\right)\sqrt{2}\right)\\
&=&\dfrac{47}{6}+\left(-\dfrac{526}{7}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{29}{3}\right)\sqrt{4}+\left(-9\right)\sqrt{50}+\left(\dfrac{67}{7}\right)\sqrt{18}+\left(-\dfrac{71}{7}\right)\sqrt{8}+\left(-\dfrac{23}{4}\right)\sqrt{4}\right)-\left(\left(-\dfrac{76}{7}\right)\sqrt{50}+0-\left(\left(-\dfrac{22}{7}\right)\sqrt{50}\right)\right)\\
&=&\left(\dfrac{58}{3}+\left(-45\right)\sqrt{2}+\left(\dfrac{201}{7}\right)\sqrt{2}+\left(-\dfrac{142}{7}\right)\sqrt{2}-\dfrac{23}{2}\right)-\left(\left(-\dfrac{380}{7}\right)\sqrt{2}+0-\left(\left(-\dfrac{110}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{47}{6}+\left(-\dfrac{256}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{270}{7}\right)\sqrt{2}+0\right)\\
&=&\dfrac{47}{6}+\left(-\dfrac{256}{7}\right)\sqrt{2}+\left(\dfrac{270}{7}\right)\sqrt{2}+0\\
&=&\dfrac{47}{6}+\left(2\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{29}{3}\right)\sqrt{4}+\left(-9\right)\sqrt{50}+\left(\dfrac{67}{7}\right)\sqrt{18}+\left(-\dfrac{71}{7}\right)\sqrt{8}+\left(-\dfrac{23}{4}\right)\sqrt{4}\right)\times\left(\left(-\dfrac{76}{7}\right)\sqrt{50}+0-\left(\left(-\dfrac{22}{7}\right)\sqrt{50}\right)\right)\\
&=&\left(\dfrac{58}{3}+\left(-45\right)\sqrt{2}+\left(\dfrac{201}{7}\right)\sqrt{2}+\left(-\dfrac{142}{7}\right)\sqrt{2}-\dfrac{23}{2}\right)\times\left(\left(-\dfrac{380}{7}\right)\sqrt{2}+0-\left(\left(-\dfrac{110}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{47}{6}+\left(-\dfrac{256}{7}\right)\sqrt{2}\right)\left(\left(-\dfrac{270}{7}\right)\sqrt{2}+0\right)\\
&=&\left(-\dfrac{2115}{7}\right)\sqrt{2}+0+\left(\dfrac{69120}{49}\right)\sqrt{4}\\
&=&\left(-\dfrac{2115}{7}\right)\sqrt{2}+\dfrac{138240}{49}\\
\end{eqnarray*}