L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-7\right)\sqrt{75}\right)-\left(\left(\dfrac{64}{5}\right)\sqrt{75}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(-\dfrac{13}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{50}{7}\right)\sqrt{12}\right)+\left(\left(\dfrac{19}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{1}{6}\right)\sqrt{9}\right)-7\) et \( Y=\left(-\dfrac{81}{5}-\dfrac{9}{4}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{23}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{13}{7}\right)\sqrt{75}\right)\right)\) . 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(-7\right)\sqrt{75}\right)-\left(\left(\dfrac{64}{5}\right)\sqrt{75}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(-\dfrac{13}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{50}{7}\right)\sqrt{12}\right)+\left(\left(\dfrac{19}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{1}{6}\right)\sqrt{9}\right)-7\right)+\left(\left(-\dfrac{81}{5}-\dfrac{9}{4}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{23}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{13}{7}\right)\sqrt{75}\right)\right)\right)\\
&=&\left(\left(\left(-35\right)\sqrt{3}\right)-\left(\left(64\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(-\dfrac{39}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{100}{7}\right)\sqrt{3}\right)+\left(\left(19\right)\sqrt{3}\right)+\dfrac{1}{2}-7\right)+\left(\left(-\dfrac{81}{5}-\dfrac{9}{4}\right)-0-\left(\left(-\dfrac{130}{7}\right)\sqrt{3}\right)-\left(\left(\left(23\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{7}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-35\right)\sqrt{3}\right)-\left(\left(64\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(-\dfrac{39}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{100}{7}\right)\sqrt{3}\right)+\left(\left(19\right)\sqrt{3}\right)+\dfrac{1}{2}-7+\left(-\dfrac{81}{5}-\dfrac{9}{4}\right)-0-\left(\left(-\dfrac{130}{7}\right)\sqrt{3}\right)-\left(\left(\left(23\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{451}{5}\right)\sqrt{3}-\dfrac{499}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-7\right)\sqrt{75}\right)-\left(\left(\dfrac{64}{5}\right)\sqrt{75}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(-\dfrac{13}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{50}{7}\right)\sqrt{12}\right)+\left(\left(\dfrac{19}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{1}{6}\right)\sqrt{9}\right)-7\right)-\left(\left(-\dfrac{81}{5}-\dfrac{9}{4}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{23}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{13}{7}\right)\sqrt{75}\right)\right)\right)\\
&=&\left(\left(\left(-35\right)\sqrt{3}\right)-\left(\left(64\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(-\dfrac{39}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{100}{7}\right)\sqrt{3}\right)+\left(\left(19\right)\sqrt{3}\right)+\dfrac{1}{2}-7\right)-\left(\left(-\dfrac{81}{5}-\dfrac{9}{4}\right)-0-\left(\left(-\dfrac{130}{7}\right)\sqrt{3}\right)-\left(\left(\left(23\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{7}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{2677}{35}\right)\sqrt{3}-\dfrac{13}{2}\right)-\left(-\dfrac{369}{20}+\left(-\dfrac{96}{7}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{2677}{35}\right)\sqrt{3}-\dfrac{13}{2}+\dfrac{369}{20}+\left(\dfrac{96}{7}\right)\sqrt{3}\\
&=&\left(-\dfrac{2197}{35}\right)\sqrt{3}+\dfrac{239}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-7\right)\sqrt{75}\right)-\left(\left(\dfrac{64}{5}\right)\sqrt{75}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(-\dfrac{13}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{50}{7}\right)\sqrt{12}\right)+\left(\left(\dfrac{19}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{1}{6}\right)\sqrt{9}\right)-7\right)\times\left(\left(-\dfrac{81}{5}-\dfrac{9}{4}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{23}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{13}{7}\right)\sqrt{75}\right)\right)\right)\\
&=&\left(\left(\left(-35\right)\sqrt{3}\right)-\left(\left(64\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(-\dfrac{39}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{100}{7}\right)\sqrt{3}\right)+\left(\left(19\right)\sqrt{3}\right)+\dfrac{1}{2}-7\right)\times\left(\left(-\dfrac{81}{5}-\dfrac{9}{4}\right)-0-\left(\left(-\dfrac{130}{7}\right)\sqrt{3}\right)-\left(\left(\left(23\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{7}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{2677}{35}\right)\sqrt{3}-\dfrac{13}{2}\right)\left(-\dfrac{369}{20}+\left(-\dfrac{96}{7}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{1050213}{700}\right)\sqrt{3}+\left(\dfrac{256992}{245}\right)\sqrt{9}+\dfrac{4797}{40}\\
&=&\left(\dfrac{1050213}{700}\right)\sqrt{3}+\dfrac{6402861}{1960}\\
\end{eqnarray*}