L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-8\right)\sqrt{75}\) et \( Y=\left(\left(-\dfrac{25}{6}\right)\sqrt{12}\right)-\left(\left(-\dfrac{81}{5}\right)\sqrt{27}\right)+\dfrac{21}{2}-\left(\left(\dfrac{46}{3}\right)\sqrt{27}\right)+\left(\dfrac{71}{9}\right)\sqrt{12}-7-\left(\left(-\dfrac{81}{7}\right)\sqrt{9}\right)-\left(\left(2\right)\sqrt{75}\right)+\left(-\dfrac{19}{5}\right)\sqrt{75}+\left(8\right)\sqrt{9}+\left(-3\right)\sqrt{27}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-8\right)\sqrt{75}\right)+\left(\left(\left(-\dfrac{25}{6}\right)\sqrt{12}\right)-\left(\left(-\dfrac{81}{5}\right)\sqrt{27}\right)+\dfrac{21}{2}-\left(\left(\dfrac{46}{3}\right)\sqrt{27}\right)+\left(\dfrac{71}{9}\right)\sqrt{12}-7-\left(\left(-\dfrac{81}{7}\right)\sqrt{9}\right)-\left(\left(2\right)\sqrt{75}\right)+\left(-\dfrac{19}{5}\right)\sqrt{75}+\left(8\right)\sqrt{9}+\left(-3\right)\sqrt{27}\right)\\
&=&\left(\left(-40\right)\sqrt{3}\right)+\left(\left(\left(-\dfrac{25}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{243}{5}\right)\sqrt{3}\right)+\dfrac{21}{2}-\left(\left(46\right)\sqrt{3}\right)+\left(\dfrac{142}{9}\right)\sqrt{3}-7+\dfrac{243}{7}-\left(\left(10\right)\sqrt{3}\right)+\left(-19\right)\sqrt{3}+24+\left(-9\right)\sqrt{3}\right)\\
&=&\left(-40\right)\sqrt{3}+\left(\left(-\dfrac{25}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{243}{5}\right)\sqrt{3}\right)+\dfrac{21}{2}-\left(\left(46\right)\sqrt{3}\right)+\left(\dfrac{142}{9}\right)\sqrt{3}-7+\dfrac{243}{7}-\left(\left(10\right)\sqrt{3}\right)+\left(-19\right)\sqrt{3}+24+\left(-9\right)\sqrt{3}\\
&=&\left(-\dfrac{3058}{45}\right)\sqrt{3}+\dfrac{871}{14}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-8\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{25}{6}\right)\sqrt{12}\right)-\left(\left(-\dfrac{81}{5}\right)\sqrt{27}\right)+\dfrac{21}{2}-\left(\left(\dfrac{46}{3}\right)\sqrt{27}\right)+\left(\dfrac{71}{9}\right)\sqrt{12}-7-\left(\left(-\dfrac{81}{7}\right)\sqrt{9}\right)-\left(\left(2\right)\sqrt{75}\right)+\left(-\dfrac{19}{5}\right)\sqrt{75}+\left(8\right)\sqrt{9}+\left(-3\right)\sqrt{27}\right)\\
&=&\left(\left(-40\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{25}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{243}{5}\right)\sqrt{3}\right)+\dfrac{21}{2}-\left(\left(46\right)\sqrt{3}\right)+\left(\dfrac{142}{9}\right)\sqrt{3}-7+\dfrac{243}{7}-\left(\left(10\right)\sqrt{3}\right)+\left(-19\right)\sqrt{3}+24+\left(-9\right)\sqrt{3}\right)\\
&=&\left(\left(-40\right)\sqrt{3}\right)-\left(\left(-\dfrac{1258}{45}\right)\sqrt{3}+\dfrac{871}{14}\right)\\
&=&\left(-40\right)\sqrt{3}+\left(\dfrac{1258}{45}\right)\sqrt{3}-\dfrac{871}{14}\\
&=&\left(-\dfrac{542}{45}\right)\sqrt{3}-\dfrac{871}{14}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-8\right)\sqrt{75}\right)\times\left(\left(\left(-\dfrac{25}{6}\right)\sqrt{12}\right)-\left(\left(-\dfrac{81}{5}\right)\sqrt{27}\right)+\dfrac{21}{2}-\left(\left(\dfrac{46}{3}\right)\sqrt{27}\right)+\left(\dfrac{71}{9}\right)\sqrt{12}-7-\left(\left(-\dfrac{81}{7}\right)\sqrt{9}\right)-\left(\left(2\right)\sqrt{75}\right)+\left(-\dfrac{19}{5}\right)\sqrt{75}+\left(8\right)\sqrt{9}+\left(-3\right)\sqrt{27}\right)\\
&=&\left(\left(-40\right)\sqrt{3}\right)\times\left(\left(\left(-\dfrac{25}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{243}{5}\right)\sqrt{3}\right)+\dfrac{21}{2}-\left(\left(46\right)\sqrt{3}\right)+\left(\dfrac{142}{9}\right)\sqrt{3}-7+\dfrac{243}{7}-\left(\left(10\right)\sqrt{3}\right)+\left(-19\right)\sqrt{3}+24+\left(-9\right)\sqrt{3}\right)\\
&=&\left(\left(-40\right)\sqrt{3}\right)\left(\left(-\dfrac{1258}{45}\right)\sqrt{3}+\dfrac{871}{14}\right)\\
&=&\left(\dfrac{10064}{9}\right)\sqrt{9}+\left(-\dfrac{17420}{7}\right)\sqrt{3}\\
&=&\dfrac{10064}{3}+\left(-\dfrac{17420}{7}\right)\sqrt{3}\\
\end{eqnarray*}