L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{41}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{12}-\dfrac{17}{7}+\left(-\dfrac{14}{5}\right)\sqrt{75}\right)\) et \( Y=\left(\left(\dfrac{57}{2}\right)\sqrt{75}\right)-\left(\left(-6\right)\sqrt{12}+\left(6\right)\sqrt{75}+\left(-8\right)\sqrt{27}\right)\) . 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(\left(\dfrac{41}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{12}-\dfrac{17}{7}+\left(-\dfrac{14}{5}\right)\sqrt{75}\right)\right)+\left(\left(\left(\dfrac{57}{2}\right)\sqrt{75}\right)-\left(\left(-6\right)\sqrt{12}+\left(6\right)\sqrt{75}+\left(-8\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\dfrac{123}{2}-\left(\left(-\dfrac{102}{5}\right)\sqrt{3}\right)+24\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{3}-\dfrac{17}{7}+\left(-14\right)\sqrt{3}\right)\right)+\left(\left(\left(\dfrac{285}{2}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}+\left(30\right)\sqrt{3}+\left(-24\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{123}{2}-\left(\left(-\dfrac{102}{5}\right)\sqrt{3}\right)+24\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{3}-\dfrac{17}{7}+\left(-14\right)\sqrt{3}\right)+\left(\left(\dfrac{285}{2}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}+\left(30\right)\sqrt{3}+\left(-24\right)\sqrt{3}\right)\\
&=&\dfrac{1231}{14}+\left(\dfrac{13523}{70}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{41}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{12}-\dfrac{17}{7}+\left(-\dfrac{14}{5}\right)\sqrt{75}\right)\right)-\left(\left(\left(\dfrac{57}{2}\right)\sqrt{75}\right)-\left(\left(-6\right)\sqrt{12}+\left(6\right)\sqrt{75}+\left(-8\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\dfrac{123}{2}-\left(\left(-\dfrac{102}{5}\right)\sqrt{3}\right)+24\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{3}-\dfrac{17}{7}+\left(-14\right)\sqrt{3}\right)\right)-\left(\left(\left(\dfrac{285}{2}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}+\left(30\right)\sqrt{3}+\left(-24\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{1231}{14}+\left(\dfrac{1564}{35}\right)\sqrt{3}\right)-\left(\left(\dfrac{297}{2}\right)\sqrt{3}\right)\\
&=&\dfrac{1231}{14}+\left(\dfrac{1564}{35}\right)\sqrt{3}+\left(-\dfrac{297}{2}\right)\sqrt{3}\\
&=&\dfrac{1231}{14}+\left(-\dfrac{7267}{70}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{41}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{12}\right)-\left(\left(-8\right)\sqrt{9}\right)\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{12}-\dfrac{17}{7}+\left(-\dfrac{14}{5}\right)\sqrt{75}\right)\right)\times\left(\left(\left(\dfrac{57}{2}\right)\sqrt{75}\right)-\left(\left(-6\right)\sqrt{12}+\left(6\right)\sqrt{75}+\left(-8\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\dfrac{123}{2}-\left(\left(-\dfrac{102}{5}\right)\sqrt{3}\right)+24\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{3}-\dfrac{17}{7}+\left(-14\right)\sqrt{3}\right)\right)\times\left(\left(\left(\dfrac{285}{2}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}+\left(30\right)\sqrt{3}+\left(-24\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{1231}{14}+\left(\dfrac{1564}{35}\right)\sqrt{3}\right)\left(\left(\dfrac{297}{2}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{365607}{28}\right)\sqrt{3}+\left(\dfrac{232254}{35}\right)\sqrt{9}\\
&=&\left(\dfrac{365607}{28}\right)\sqrt{3}+\dfrac{696762}{35}\\
\end{eqnarray*}