L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-4\right)\sqrt{8}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{50}\right)+\left(-6\right)\sqrt{18}+\left(\dfrac{79}{2}\right)\sqrt{18}+\left(\left(-\dfrac{65}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{3}{8}\right)\sqrt{8}\right)-\left(\left(-\dfrac{8}{7}\right)\sqrt{8}\right)\) et \( Y=\left(\left(\dfrac{48}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{62}{5}\right)\sqrt{8}+5+\left(\dfrac{21}{8}\right)\sqrt{4}+\left(\dfrac{13}{6}\right)\sqrt{18}+\left(-\dfrac{41}{2}\right)\sqrt{4}\right)\) . 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(-4\right)\sqrt{8}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{50}\right)+\left(-6\right)\sqrt{18}+\left(\dfrac{79}{2}\right)\sqrt{18}+\left(\left(-\dfrac{65}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{3}{8}\right)\sqrt{8}\right)-\left(\left(-\dfrac{8}{7}\right)\sqrt{8}\right)\right)+\left(\left(\left(\dfrac{48}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{62}{5}\right)\sqrt{8}+5+\left(\dfrac{21}{8}\right)\sqrt{4}+\left(\dfrac{13}{6}\right)\sqrt{18}+\left(-\dfrac{41}{2}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\left(-8\right)\sqrt{2}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{2}\right)+\left(-18\right)\sqrt{2}+\left(\dfrac{237}{2}\right)\sqrt{2}+\left(\left(-\dfrac{130}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{16}{7}\right)\sqrt{2}\right)\right)+\left(\left(\left(\dfrac{144}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{5}\right)\sqrt{2}+5+\dfrac{21}{4}+\left(\dfrac{13}{2}\right)\sqrt{2}-41\right)\right)\\
&=&\left(\left(-8\right)\sqrt{2}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{2}\right)+\left(-18\right)\sqrt{2}+\left(\dfrac{237}{2}\right)\sqrt{2}+\left(\left(-\dfrac{130}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{16}{7}\right)\sqrt{2}\right)+\left(\left(\dfrac{144}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{5}\right)\sqrt{2}+5+\dfrac{21}{4}+\left(\dfrac{13}{2}\right)\sqrt{2}-41\right)\\
&=&\left(\dfrac{16655}{252}\right)\sqrt{2}+\dfrac{123}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-4\right)\sqrt{8}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{50}\right)+\left(-6\right)\sqrt{18}+\left(\dfrac{79}{2}\right)\sqrt{18}+\left(\left(-\dfrac{65}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{3}{8}\right)\sqrt{8}\right)-\left(\left(-\dfrac{8}{7}\right)\sqrt{8}\right)\right)-\left(\left(\left(\dfrac{48}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{62}{5}\right)\sqrt{8}+5+\left(\dfrac{21}{8}\right)\sqrt{4}+\left(\dfrac{13}{6}\right)\sqrt{18}+\left(-\dfrac{41}{2}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\left(-8\right)\sqrt{2}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{2}\right)+\left(-18\right)\sqrt{2}+\left(\dfrac{237}{2}\right)\sqrt{2}+\left(\left(-\dfrac{130}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{16}{7}\right)\sqrt{2}\right)\right)-\left(\left(\left(\dfrac{144}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{5}\right)\sqrt{2}+5+\dfrac{21}{4}+\left(\dfrac{13}{2}\right)\sqrt{2}-41\right)\right)\\
&=&\left(\left(\dfrac{17285}{252}\right)\sqrt{2}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{2}+\dfrac{123}{4}\right)\\
&=&\left(\dfrac{17285}{252}\right)\sqrt{2}+\left(\dfrac{5}{2}\right)\sqrt{2}-\dfrac{123}{4}\\
&=&\left(\dfrac{17915}{252}\right)\sqrt{2}-\dfrac{123}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-4\right)\sqrt{8}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{50}\right)+\left(-6\right)\sqrt{18}+\left(\dfrac{79}{2}\right)\sqrt{18}+\left(\left(-\dfrac{65}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{3}{8}\right)\sqrt{8}\right)-\left(\left(-\dfrac{8}{7}\right)\sqrt{8}\right)\right)\times\left(\left(\left(\dfrac{48}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{62}{5}\right)\sqrt{8}+5+\left(\dfrac{21}{8}\right)\sqrt{4}+\left(\dfrac{13}{6}\right)\sqrt{18}+\left(-\dfrac{41}{2}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\left(-8\right)\sqrt{2}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{2}\right)+\left(-18\right)\sqrt{2}+\left(\dfrac{237}{2}\right)\sqrt{2}+\left(\left(-\dfrac{130}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{16}{7}\right)\sqrt{2}\right)\right)\times\left(\left(\left(\dfrac{144}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{5}\right)\sqrt{2}+5+\dfrac{21}{4}+\left(\dfrac{13}{2}\right)\sqrt{2}-41\right)\right)\\
&=&\left(\left(\dfrac{17285}{252}\right)\sqrt{2}\right)\left(\left(-\dfrac{5}{2}\right)\sqrt{2}+\dfrac{123}{4}\right)\\
&=&\left(-\dfrac{86425}{504}\right)\sqrt{4}+\left(\dfrac{708685}{336}\right)\sqrt{2}\\
&=&-\dfrac{86425}{252}+\left(\dfrac{708685}{336}\right)\sqrt{2}\\
\end{eqnarray*}