L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{25}{4}\right)\sqrt{18}+\left(9\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{25}{4}\right)\sqrt{18}\right)+\dfrac{48}{7}-\left(\left(-1\right)\sqrt{8}\right)\right)-\left(\left(1\right)\sqrt{8}\right)\) et \( Y=\left(\dfrac{61}{5}+\left(\dfrac{23}{6}\right)\sqrt{4}\right)-\left(-\dfrac{7}{4}+\left(\dfrac{71}{3}\right)\sqrt{18}+\left(\dfrac{11}{7}\right)\sqrt{18}+\left(6\right)\sqrt{18}\right)\) . 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{25}{4}\right)\sqrt{18}+\left(9\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{25}{4}\right)\sqrt{18}\right)+\dfrac{48}{7}-\left(\left(-1\right)\sqrt{8}\right)\right)-\left(\left(1\right)\sqrt{8}\right)\right)+\left(\left(\dfrac{61}{5}+\left(\dfrac{23}{6}\right)\sqrt{4}\right)-\left(-\dfrac{7}{4}+\left(\dfrac{71}{3}\right)\sqrt{18}+\left(\dfrac{11}{7}\right)\sqrt{18}+\left(6\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{2}+18\right)-\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{2}\right)+\dfrac{48}{7}-\left(\left(-2\right)\sqrt{2}\right)\right)-\left(\left(2\right)\sqrt{2}\right)\right)+\left(\left(\dfrac{61}{5}+\dfrac{23}{3}\right)-\left(-\dfrac{7}{4}+\left(71\right)\sqrt{2}+\left(\dfrac{33}{7}\right)\sqrt{2}+\left(18\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{75}{4}\right)\sqrt{2}+18\right)-\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{2}\right)+\dfrac{48}{7}-\left(\left(-2\right)\sqrt{2}\right)\right)-\left(\left(2\right)\sqrt{2}\right)+\left(\dfrac{61}{5}+\dfrac{23}{3}\right)-\left(-\dfrac{7}{4}+\left(71\right)\sqrt{2}+\left(\dfrac{33}{7}\right)\sqrt{2}+\left(18\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{684}{7}\right)\sqrt{2}+\dfrac{13759}{420}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{25}{4}\right)\sqrt{18}+\left(9\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{25}{4}\right)\sqrt{18}\right)+\dfrac{48}{7}-\left(\left(-1\right)\sqrt{8}\right)\right)-\left(\left(1\right)\sqrt{8}\right)\right)-\left(\left(\dfrac{61}{5}+\left(\dfrac{23}{6}\right)\sqrt{4}\right)-\left(-\dfrac{7}{4}+\left(\dfrac{71}{3}\right)\sqrt{18}+\left(\dfrac{11}{7}\right)\sqrt{18}+\left(6\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{2}+18\right)-\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{2}\right)+\dfrac{48}{7}-\left(\left(-2\right)\sqrt{2}\right)\right)-\left(\left(2\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{61}{5}+\dfrac{23}{3}\right)-\left(-\dfrac{7}{4}+\left(71\right)\sqrt{2}+\left(\dfrac{33}{7}\right)\sqrt{2}+\left(18\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-4\right)\sqrt{2}+\dfrac{78}{7}\right)-\left(\dfrac{1297}{60}+\left(-\dfrac{656}{7}\right)\sqrt{2}\right)\\
&=&\left(-4\right)\sqrt{2}+\dfrac{78}{7}+-\dfrac{1297}{60}+\left(\dfrac{656}{7}\right)\sqrt{2}\\
&=&\left(\dfrac{628}{7}\right)\sqrt{2}-\dfrac{4399}{420}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{25}{4}\right)\sqrt{18}+\left(9\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{25}{4}\right)\sqrt{18}\right)+\dfrac{48}{7}-\left(\left(-1\right)\sqrt{8}\right)\right)-\left(\left(1\right)\sqrt{8}\right)\right)\times\left(\left(\dfrac{61}{5}+\left(\dfrac{23}{6}\right)\sqrt{4}\right)-\left(-\dfrac{7}{4}+\left(\dfrac{71}{3}\right)\sqrt{18}+\left(\dfrac{11}{7}\right)\sqrt{18}+\left(6\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{2}+18\right)-\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{2}\right)+\dfrac{48}{7}-\left(\left(-2\right)\sqrt{2}\right)\right)-\left(\left(2\right)\sqrt{2}\right)\right)\times\left(\left(\dfrac{61}{5}+\dfrac{23}{3}\right)-\left(-\dfrac{7}{4}+\left(71\right)\sqrt{2}+\left(\dfrac{33}{7}\right)\sqrt{2}+\left(18\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-4\right)\sqrt{2}+\dfrac{78}{7}\right)\left(\dfrac{1297}{60}+\left(-\dfrac{656}{7}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{831073}{735}\right)\sqrt{2}+\left(\dfrac{2624}{7}\right)\sqrt{4}+\dfrac{16861}{70}\\
&=&\left(-\dfrac{831073}{735}\right)\sqrt{2}+\dfrac{69341}{70}\\
\end{eqnarray*}