L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{76}{7}\right)\sqrt{27}+\dfrac{5}{9}+0\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{27}\right)\) et \( Y=\dfrac{25}{4}-\left(\left(\dfrac{45}{4}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{43}{4}\right)\sqrt{12}\right)-\left(\left(3\right)\sqrt{12}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{27}\right)\right)\) . 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(-\dfrac{76}{7}\right)\sqrt{27}+\dfrac{5}{9}+0\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{27}\right)\right)+\left(\dfrac{25}{4}-\left(\left(\dfrac{45}{4}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{43}{4}\right)\sqrt{12}\right)-\left(\left(3\right)\sqrt{12}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{27}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{228}{7}\right)\sqrt{3}+\dfrac{5}{9}+0\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{3}\right)\right)+\left(\dfrac{25}{4}-\left(\left(\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)-\left(\left(\dfrac{123}{4}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{228}{7}\right)\sqrt{3}+\dfrac{5}{9}+0\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{3}\right)+\dfrac{25}{4}-\left(\left(\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)-\left(\left(\dfrac{123}{4}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{457}{28}\right)\sqrt{3}+\dfrac{245}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{76}{7}\right)\sqrt{27}+\dfrac{5}{9}+0\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{27}\right)\right)-\left(\dfrac{25}{4}-\left(\left(\dfrac{45}{4}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{43}{4}\right)\sqrt{12}\right)-\left(\left(3\right)\sqrt{12}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{27}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{228}{7}\right)\sqrt{3}+\dfrac{5}{9}+0\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{3}\right)\right)-\left(\dfrac{25}{4}-\left(\left(\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)-\left(\left(\dfrac{123}{4}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{729}{14}\right)\sqrt{3}+\dfrac{5}{9}\right)-\left(\dfrac{25}{4}+\left(\dfrac{143}{4}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{729}{14}\right)\sqrt{3}+\dfrac{5}{9}+-\dfrac{25}{4}+\left(-\dfrac{143}{4}\right)\sqrt{3}\\
&=&\left(-\dfrac{2459}{28}\right)\sqrt{3}-\dfrac{205}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{76}{7}\right)\sqrt{27}+\dfrac{5}{9}+0\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{27}\right)\right)\times\left(\dfrac{25}{4}-\left(\left(\dfrac{45}{4}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{43}{4}\right)\sqrt{12}\right)-\left(\left(3\right)\sqrt{12}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{27}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{228}{7}\right)\sqrt{3}+\dfrac{5}{9}+0\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{3}\right)\right)\times\left(\dfrac{25}{4}-\left(\left(\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)-\left(\left(\dfrac{123}{4}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{729}{14}\right)\sqrt{3}+\dfrac{5}{9}\right)\left(\dfrac{25}{4}+\left(\dfrac{143}{4}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{154015}{504}\right)\sqrt{3}+\left(-\dfrac{104247}{56}\right)\sqrt{9}+\dfrac{125}{36}\\
&=&\left(-\dfrac{154015}{504}\right)\sqrt{3}-\dfrac{2812919}{504}\\
\end{eqnarray*}