L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=2+\dfrac{8}{5}+\left(\dfrac{25}{4}\right)\sqrt{18}+\left(7\right)\sqrt{8}+\left(\dfrac{65}{9}\right)\sqrt{50}+\left(-\dfrac{79}{7}\right)\sqrt{50}+\left(\left(-\dfrac{81}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{14}{5}\right)\sqrt{50}\right)+\left(-\dfrac{19}{3}\right)\sqrt{50}+\left(-\dfrac{48}{7}\right)\sqrt{18}+\left(-\dfrac{20}{3}\right)\sqrt{4}+\left(-\dfrac{71}{2}\right)\sqrt{8}+0\) et \( Y=\left(-\dfrac{41}{4}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(2+\dfrac{8}{5}+\left(\dfrac{25}{4}\right)\sqrt{18}+\left(7\right)\sqrt{8}+\left(\dfrac{65}{9}\right)\sqrt{50}+\left(-\dfrac{79}{7}\right)\sqrt{50}+\left(\left(-\dfrac{81}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{14}{5}\right)\sqrt{50}\right)+\left(-\dfrac{19}{3}\right)\sqrt{50}+\left(-\dfrac{48}{7}\right)\sqrt{18}+\left(-\dfrac{20}{3}\right)\sqrt{4}+\left(-\dfrac{71}{2}\right)\sqrt{8}+0\right)+\left(\left(-\dfrac{41}{4}\right)\sqrt{18}\right)\\
&=&\left(2+\dfrac{8}{5}+\left(\dfrac{75}{4}\right)\sqrt{2}+\left(14\right)\sqrt{2}+\left(\dfrac{325}{9}\right)\sqrt{2}+\left(-\dfrac{395}{7}\right)\sqrt{2}+\left(\left(-\dfrac{405}{4}\right)\sqrt{2}\right)-\left(\left(14\right)\sqrt{2}\right)+\left(-\dfrac{95}{3}\right)\sqrt{2}+\left(-\dfrac{144}{7}\right)\sqrt{2}-\dfrac{40}{3}+\left(-71\right)\sqrt{2}+0\right)+\left(\left(-\dfrac{123}{4}\right)\sqrt{2}\right)\\
&=&2+\dfrac{8}{5}+\left(\dfrac{75}{4}\right)\sqrt{2}+\left(14\right)\sqrt{2}+\left(\dfrac{325}{9}\right)\sqrt{2}+\left(-\dfrac{395}{7}\right)\sqrt{2}+\left(\left(-\dfrac{405}{4}\right)\sqrt{2}\right)-\left(\left(14\right)\sqrt{2}\right)+\left(-\dfrac{95}{3}\right)\sqrt{2}+\left(-\dfrac{144}{7}\right)\sqrt{2}-\dfrac{40}{3}+\left(-71\right)\sqrt{2}+0+\left(-\dfrac{123}{4}\right)\sqrt{2}\\
&=&-\dfrac{146}{15}+\left(-\dfrac{9245}{36}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(2+\dfrac{8}{5}+\left(\dfrac{25}{4}\right)\sqrt{18}+\left(7\right)\sqrt{8}+\left(\dfrac{65}{9}\right)\sqrt{50}+\left(-\dfrac{79}{7}\right)\sqrt{50}+\left(\left(-\dfrac{81}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{14}{5}\right)\sqrt{50}\right)+\left(-\dfrac{19}{3}\right)\sqrt{50}+\left(-\dfrac{48}{7}\right)\sqrt{18}+\left(-\dfrac{20}{3}\right)\sqrt{4}+\left(-\dfrac{71}{2}\right)\sqrt{8}+0\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{18}\right)\\
&=&\left(2+\dfrac{8}{5}+\left(\dfrac{75}{4}\right)\sqrt{2}+\left(14\right)\sqrt{2}+\left(\dfrac{325}{9}\right)\sqrt{2}+\left(-\dfrac{395}{7}\right)\sqrt{2}+\left(\left(-\dfrac{405}{4}\right)\sqrt{2}\right)-\left(\left(14\right)\sqrt{2}\right)+\left(-\dfrac{95}{3}\right)\sqrt{2}+\left(-\dfrac{144}{7}\right)\sqrt{2}-\dfrac{40}{3}+\left(-71\right)\sqrt{2}+0\right)-\left(\left(-\dfrac{123}{4}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{146}{15}+\left(-\dfrac{4069}{18}\right)\sqrt{2}\right)-\left(\left(-\dfrac{123}{4}\right)\sqrt{2}\right)\\
&=&-\dfrac{146}{15}+\left(-\dfrac{4069}{18}\right)\sqrt{2}+\left(\dfrac{123}{4}\right)\sqrt{2}\\
&=&-\dfrac{146}{15}+\left(-\dfrac{7031}{36}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(2+\dfrac{8}{5}+\left(\dfrac{25}{4}\right)\sqrt{18}+\left(7\right)\sqrt{8}+\left(\dfrac{65}{9}\right)\sqrt{50}+\left(-\dfrac{79}{7}\right)\sqrt{50}+\left(\left(-\dfrac{81}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{14}{5}\right)\sqrt{50}\right)+\left(-\dfrac{19}{3}\right)\sqrt{50}+\left(-\dfrac{48}{7}\right)\sqrt{18}+\left(-\dfrac{20}{3}\right)\sqrt{4}+\left(-\dfrac{71}{2}\right)\sqrt{8}+0\right)\times\left(\left(-\dfrac{41}{4}\right)\sqrt{18}\right)\\
&=&\left(2+\dfrac{8}{5}+\left(\dfrac{75}{4}\right)\sqrt{2}+\left(14\right)\sqrt{2}+\left(\dfrac{325}{9}\right)\sqrt{2}+\left(-\dfrac{395}{7}\right)\sqrt{2}+\left(\left(-\dfrac{405}{4}\right)\sqrt{2}\right)-\left(\left(14\right)\sqrt{2}\right)+\left(-\dfrac{95}{3}\right)\sqrt{2}+\left(-\dfrac{144}{7}\right)\sqrt{2}-\dfrac{40}{3}+\left(-71\right)\sqrt{2}+0\right)\times\left(\left(-\dfrac{123}{4}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{146}{15}+\left(-\dfrac{4069}{18}\right)\sqrt{2}\right)\left(\left(-\dfrac{123}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2993}{10}\right)\sqrt{2}+\left(\dfrac{166829}{24}\right)\sqrt{4}\\
&=&\left(\dfrac{2993}{10}\right)\sqrt{2}+\dfrac{166829}{12}\\
\end{eqnarray*}