L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{63}{5}\right)\sqrt{4}\right)-\left(\dfrac{4}{3}+\left(\dfrac{1}{3}\right)\sqrt{4}+\left(\dfrac{22}{7}\right)\sqrt{50}+\left(\dfrac{8}{7}\right)\sqrt{8}-9\right)-\left(\left(\left(\dfrac{36}{7}\right)\sqrt{4}\right)+9\right)-\left(\left(\left(-\dfrac{77}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{8}\right)-5\right)-\left(\left(\left(-\dfrac{1}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{8}\right)\right)\) et \( Y=\left(-5\right)\sqrt{4}\) . 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{63}{5}\right)\sqrt{4}\right)-\left(\dfrac{4}{3}+\left(\dfrac{1}{3}\right)\sqrt{4}+\left(\dfrac{22}{7}\right)\sqrt{50}+\left(\dfrac{8}{7}\right)\sqrt{8}-9\right)-\left(\left(\left(\dfrac{36}{7}\right)\sqrt{4}\right)+9\right)-\left(\left(\left(-\dfrac{77}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{8}\right)-5\right)-\left(\left(\left(-\dfrac{1}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{8}\right)\right)\right)+\left(\left(-5\right)\sqrt{4}\right)\\
&=&\left(\dfrac{126}{5}-\left(\dfrac{4}{3}+\dfrac{2}{3}+\left(\dfrac{110}{7}\right)\sqrt{2}+\left(\dfrac{16}{7}\right)\sqrt{2}-9\right)-\left(\dfrac{72}{7}+9\right)-\left(\left(\left(-\dfrac{231}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{39}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{96}{5}\right)\sqrt{2}\right)-5\right)-\left(-\dfrac{1}{2}-\left(\left(\dfrac{31}{2}\right)\sqrt{2}\right)\right)\right)+\left(-10\right)\\
&=&\dfrac{126}{5}-\left(\dfrac{4}{3}+\dfrac{2}{3}+\left(\dfrac{110}{7}\right)\sqrt{2}+\left(\dfrac{16}{7}\right)\sqrt{2}-9\right)-\left(\dfrac{72}{7}+9\right)-\left(\left(\left(-\dfrac{231}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{39}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{96}{5}\right)\sqrt{2}\right)-5\right)-\left(-\dfrac{1}{2}-\left(\left(\dfrac{31}{2}\right)\sqrt{2}\right)\right)-10\\
&=&\dfrac{589}{70}+\left(\dfrac{3947}{40}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{63}{5}\right)\sqrt{4}\right)-\left(\dfrac{4}{3}+\left(\dfrac{1}{3}\right)\sqrt{4}+\left(\dfrac{22}{7}\right)\sqrt{50}+\left(\dfrac{8}{7}\right)\sqrt{8}-9\right)-\left(\left(\left(\dfrac{36}{7}\right)\sqrt{4}\right)+9\right)-\left(\left(\left(-\dfrac{77}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{8}\right)-5\right)-\left(\left(\left(-\dfrac{1}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{8}\right)\right)\right)-\left(\left(-5\right)\sqrt{4}\right)\\
&=&\left(\dfrac{126}{5}-\left(\dfrac{4}{3}+\dfrac{2}{3}+\left(\dfrac{110}{7}\right)\sqrt{2}+\left(\dfrac{16}{7}\right)\sqrt{2}-9\right)-\left(\dfrac{72}{7}+9\right)-\left(\left(\left(-\dfrac{231}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{39}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{96}{5}\right)\sqrt{2}\right)-5\right)-\left(-\dfrac{1}{2}-\left(\left(\dfrac{31}{2}\right)\sqrt{2}\right)\right)\right)-\left(-10\right)\\
&=&\left(\dfrac{1289}{70}+\left(\dfrac{3947}{40}\right)\sqrt{2}\right)-\left(-10\right)\\
&=&\dfrac{1289}{70}+\left(\dfrac{3947}{40}\right)\sqrt{2}+10\\
&=&\dfrac{1989}{70}+\left(\dfrac{3947}{40}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{63}{5}\right)\sqrt{4}\right)-\left(\dfrac{4}{3}+\left(\dfrac{1}{3}\right)\sqrt{4}+\left(\dfrac{22}{7}\right)\sqrt{50}+\left(\dfrac{8}{7}\right)\sqrt{8}-9\right)-\left(\left(\left(\dfrac{36}{7}\right)\sqrt{4}\right)+9\right)-\left(\left(\left(-\dfrac{77}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{8}\right)-5\right)-\left(\left(\left(-\dfrac{1}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{8}\right)\right)\right)\times\left(\left(-5\right)\sqrt{4}\right)\\
&=&\left(\dfrac{126}{5}-\left(\dfrac{4}{3}+\dfrac{2}{3}+\left(\dfrac{110}{7}\right)\sqrt{2}+\left(\dfrac{16}{7}\right)\sqrt{2}-9\right)-\left(\dfrac{72}{7}+9\right)-\left(\left(\left(-\dfrac{231}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{39}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{96}{5}\right)\sqrt{2}\right)-5\right)-\left(-\dfrac{1}{2}-\left(\left(\dfrac{31}{2}\right)\sqrt{2}\right)\right)\right)\times\left(-10\right)\\
&=&\left(\dfrac{1289}{70}+\left(\dfrac{3947}{40}\right)\sqrt{2}\right)\left(-10\right)\\
&=&-\dfrac{1289}{7}+\left(-\dfrac{3947}{4}\right)\sqrt{2}\\
&=&-\dfrac{1289}{7}+\left(-\dfrac{3947}{4}\right)\sqrt{2}\\
\end{eqnarray*}