L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-3\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{50}\right)+\left(-3\right)\sqrt{50}+\left(-8\right)\sqrt{50}+\left(7\right)\sqrt{50}\) et \( Y=\left(\left(\dfrac{52}{9}\right)\sqrt{18}\right)+\dfrac{7}{6}-\dfrac{1}{3}+\left(-\dfrac{7}{3}\right)\sqrt{8}+\left(\left(\dfrac{34}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{31}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{9}\right)\sqrt{8}\right)\) . 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(-3\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{50}\right)+\left(-3\right)\sqrt{50}+\left(-8\right)\sqrt{50}+\left(7\right)\sqrt{50}\right)+\left(\left(\left(\dfrac{52}{9}\right)\sqrt{18}\right)+\dfrac{7}{6}-\dfrac{1}{3}+\left(-\dfrac{7}{3}\right)\sqrt{8}+\left(\left(\dfrac{34}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{31}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{9}\right)\sqrt{8}\right)\right)\\
&=&\left(-6-\left(\left(\dfrac{295}{6}\right)\sqrt{2}\right)+\left(-15\right)\sqrt{2}+\left(-40\right)\sqrt{2}+\left(35\right)\sqrt{2}\right)+\left(\left(\left(\dfrac{52}{3}\right)\sqrt{2}\right)+\dfrac{7}{6}-\dfrac{1}{3}+\left(-\dfrac{14}{3}\right)\sqrt{2}+\left(\left(34\right)\sqrt{2}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{9}\right)\sqrt{2}\right)\right)\\
&=&-6-\left(\left(\dfrac{295}{6}\right)\sqrt{2}\right)+\left(-15\right)\sqrt{2}+\left(-40\right)\sqrt{2}+\left(35\right)\sqrt{2}+\left(\left(\dfrac{52}{3}\right)\sqrt{2}\right)+\dfrac{7}{6}-\dfrac{1}{3}+\left(-\dfrac{14}{3}\right)\sqrt{2}+\left(\left(34\right)\sqrt{2}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{9}\right)\sqrt{2}\right)\\
&=&-\dfrac{31}{6}+\left(-\dfrac{443}{18}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-3\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{50}\right)+\left(-3\right)\sqrt{50}+\left(-8\right)\sqrt{50}+\left(7\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{52}{9}\right)\sqrt{18}\right)+\dfrac{7}{6}-\dfrac{1}{3}+\left(-\dfrac{7}{3}\right)\sqrt{8}+\left(\left(\dfrac{34}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{31}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{9}\right)\sqrt{8}\right)\right)\\
&=&\left(-6-\left(\left(\dfrac{295}{6}\right)\sqrt{2}\right)+\left(-15\right)\sqrt{2}+\left(-40\right)\sqrt{2}+\left(35\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{52}{3}\right)\sqrt{2}\right)+\dfrac{7}{6}-\dfrac{1}{3}+\left(-\dfrac{14}{3}\right)\sqrt{2}+\left(\left(34\right)\sqrt{2}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{9}\right)\sqrt{2}\right)\right)\\
&=&\left(-6+\left(-\dfrac{415}{6}\right)\sqrt{2}\right)-\left(\left(\dfrac{401}{9}\right)\sqrt{2}+\dfrac{5}{6}\right)\\
&=&-6+\left(-\dfrac{415}{6}\right)\sqrt{2}+\left(-\dfrac{401}{9}\right)\sqrt{2}-\dfrac{5}{6}\\
&=&-\dfrac{41}{6}+\left(-\dfrac{2047}{18}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-3\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{50}\right)+\left(-3\right)\sqrt{50}+\left(-8\right)\sqrt{50}+\left(7\right)\sqrt{50}\right)\times\left(\left(\left(\dfrac{52}{9}\right)\sqrt{18}\right)+\dfrac{7}{6}-\dfrac{1}{3}+\left(-\dfrac{7}{3}\right)\sqrt{8}+\left(\left(\dfrac{34}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{31}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{9}\right)\sqrt{8}\right)\right)\\
&=&\left(-6-\left(\left(\dfrac{295}{6}\right)\sqrt{2}\right)+\left(-15\right)\sqrt{2}+\left(-40\right)\sqrt{2}+\left(35\right)\sqrt{2}\right)\times\left(\left(\left(\dfrac{52}{3}\right)\sqrt{2}\right)+\dfrac{7}{6}-\dfrac{1}{3}+\left(-\dfrac{14}{3}\right)\sqrt{2}+\left(\left(34\right)\sqrt{2}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{9}\right)\sqrt{2}\right)\right)\\
&=&\left(-6+\left(-\dfrac{415}{6}\right)\sqrt{2}\right)\left(\left(\dfrac{401}{9}\right)\sqrt{2}+\dfrac{5}{6}\right)\\
&=&\left(-\dfrac{11699}{36}\right)\sqrt{2}-5+\left(-\dfrac{166415}{54}\right)\sqrt{4}\\
&=&\left(-\dfrac{11699}{36}\right)\sqrt{2}-\dfrac{166550}{27}\\
\end{eqnarray*}