L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{72}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{12}\right)\) et \( Y=\left(\dfrac{1}{7}-\left(\left(-4\right)\sqrt{9}\right)-\left(\left(\dfrac{29}{7}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{12}+3\right)\) . 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{72}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{12}\right)\right)+\left(\left(\dfrac{1}{7}-\left(\left(-4\right)\sqrt{9}\right)-\left(\left(\dfrac{29}{7}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{12}+3\right)\right)\\
&=&\left(\left(\left(\dfrac{144}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{96}{5}\right)\sqrt{3}\right)-\left(\left(5\right)\sqrt{3}\right)\right)+\left(\left(\dfrac{1}{7}+12-\left(\left(\dfrac{145}{7}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{3}+3\right)\right)\\
&=&\left(\left(\dfrac{144}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{96}{5}\right)\sqrt{3}\right)-\left(\left(5\right)\sqrt{3}\right)+\left(\dfrac{1}{7}+12-\left(\left(\dfrac{145}{7}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{3}+3\right)\\
&=&\left(\dfrac{96}{5}\right)\sqrt{3}+\dfrac{64}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{72}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{12}\right)\right)-\left(\left(\dfrac{1}{7}-\left(\left(-4\right)\sqrt{9}\right)-\left(\left(\dfrac{29}{7}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{12}+3\right)\right)\\
&=&\left(\left(\left(\dfrac{144}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{96}{5}\right)\sqrt{3}\right)-\left(\left(5\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{1}{7}+12-\left(\left(\dfrac{145}{7}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{3}+3\right)\right)\\
&=&\left(\left(\dfrac{1217}{35}\right)\sqrt{3}\right)-\left(\dfrac{64}{7}+\left(-\dfrac{109}{7}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{1217}{35}\right)\sqrt{3}+-\dfrac{64}{7}+\left(\dfrac{109}{7}\right)\sqrt{3}\\
&=&\left(\dfrac{1762}{35}\right)\sqrt{3}-\dfrac{64}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{72}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{12}\right)\right)\times\left(\left(\dfrac{1}{7}-\left(\left(-4\right)\sqrt{9}\right)-\left(\left(\dfrac{29}{7}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{12}+3\right)\right)\\
&=&\left(\left(\left(\dfrac{144}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{96}{5}\right)\sqrt{3}\right)-\left(\left(5\right)\sqrt{3}\right)\right)\times\left(\left(\dfrac{1}{7}+12-\left(\left(\dfrac{145}{7}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{3}+3\right)\right)\\
&=&\left(\left(\dfrac{1217}{35}\right)\sqrt{3}\right)\left(\dfrac{64}{7}+\left(-\dfrac{109}{7}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{77888}{245}\right)\sqrt{3}+\left(-\dfrac{132653}{245}\right)\sqrt{9}\\
&=&\left(\dfrac{77888}{245}\right)\sqrt{3}-\dfrac{397959}{245}\\
\end{eqnarray*}