L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(-\dfrac{17}{5}\right)\sqrt{125}+\dfrac{51}{8}+\left(-\dfrac{5}{3}\right)\sqrt{45}+\left(-\dfrac{15}{8}\right)\sqrt{25}+\left(-\dfrac{16}{9}\right)\sqrt{25}+\left(-\dfrac{79}{4}\right)\sqrt{125}+\left(\dfrac{26}{9}\right)\sqrt{20}+\left(\dfrac{17}{4}\right)\sqrt{20}+\left(\dfrac{17}{2}\right)\sqrt{20}+\left(-\dfrac{23}{2}\right)\sqrt{20}+4\) et \( Y=\left(\left(\dfrac{31}{6}\right)\sqrt{125}\right)-\left(\left(\dfrac{18}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{39}{8}\right)\sqrt{20}\right)-\left(\left(5\right)\sqrt{45}\right)-\left(\left(\dfrac{9}{5}\right)\sqrt{25}\right)+\left(-\dfrac{41}{2}\right)\sqrt{45}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{17}{5}\right)\sqrt{125}+\dfrac{51}{8}+\left(-\dfrac{5}{3}\right)\sqrt{45}+\left(-\dfrac{15}{8}\right)\sqrt{25}+\left(-\dfrac{16}{9}\right)\sqrt{25}+\left(-\dfrac{79}{4}\right)\sqrt{125}+\left(\dfrac{26}{9}\right)\sqrt{20}+\left(\dfrac{17}{4}\right)\sqrt{20}+\left(\dfrac{17}{2}\right)\sqrt{20}+\left(-\dfrac{23}{2}\right)\sqrt{20}+4\right)+\left(\left(\left(\dfrac{31}{6}\right)\sqrt{125}\right)-\left(\left(\dfrac{18}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{39}{8}\right)\sqrt{20}\right)-\left(\left(5\right)\sqrt{45}\right)-\left(\left(\dfrac{9}{5}\right)\sqrt{25}\right)+\left(-\dfrac{41}{2}\right)\sqrt{45}\right)\\
&=&\left(\left(-17\right)\sqrt{5}+\dfrac{51}{8}+\left(-5\right)\sqrt{5}-\dfrac{75}{8}-\dfrac{80}{9}+\left(-\dfrac{395}{4}\right)\sqrt{5}+\left(\dfrac{52}{9}\right)\sqrt{5}+\left(\dfrac{17}{2}\right)\sqrt{5}+\left(17\right)\sqrt{5}+\left(-23\right)\sqrt{5}+4\right)+\left(\left(\left(\dfrac{155}{6}\right)\sqrt{5}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{39}{4}\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)-9+\left(-\dfrac{123}{2}\right)\sqrt{5}\right)\\
&=&\left(-17\right)\sqrt{5}+\dfrac{51}{8}+\left(-5\right)\sqrt{5}-\dfrac{75}{8}-\dfrac{80}{9}+\left(-\dfrac{395}{4}\right)\sqrt{5}+\left(\dfrac{52}{9}\right)\sqrt{5}+\left(\dfrac{17}{2}\right)\sqrt{5}+\left(17\right)\sqrt{5}+\left(-23\right)\sqrt{5}+4+\left(\left(\dfrac{155}{6}\right)\sqrt{5}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{39}{4}\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)-9+\left(-\dfrac{123}{2}\right)\sqrt{5}\\
&=&\left(-\dfrac{11216}{63}\right)\sqrt{5}-\dfrac{152}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{17}{5}\right)\sqrt{125}+\dfrac{51}{8}+\left(-\dfrac{5}{3}\right)\sqrt{45}+\left(-\dfrac{15}{8}\right)\sqrt{25}+\left(-\dfrac{16}{9}\right)\sqrt{25}+\left(-\dfrac{79}{4}\right)\sqrt{125}+\left(\dfrac{26}{9}\right)\sqrt{20}+\left(\dfrac{17}{4}\right)\sqrt{20}+\left(\dfrac{17}{2}\right)\sqrt{20}+\left(-\dfrac{23}{2}\right)\sqrt{20}+4\right)-\left(\left(\left(\dfrac{31}{6}\right)\sqrt{125}\right)-\left(\left(\dfrac{18}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{39}{8}\right)\sqrt{20}\right)-\left(\left(5\right)\sqrt{45}\right)-\left(\left(\dfrac{9}{5}\right)\sqrt{25}\right)+\left(-\dfrac{41}{2}\right)\sqrt{45}\right)\\
&=&\left(\left(-17\right)\sqrt{5}+\dfrac{51}{8}+\left(-5\right)\sqrt{5}-\dfrac{75}{8}-\dfrac{80}{9}+\left(-\dfrac{395}{4}\right)\sqrt{5}+\left(\dfrac{52}{9}\right)\sqrt{5}+\left(\dfrac{17}{2}\right)\sqrt{5}+\left(17\right)\sqrt{5}+\left(-23\right)\sqrt{5}+4\right)-\left(\left(\left(\dfrac{155}{6}\right)\sqrt{5}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{39}{4}\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)-9+\left(-\dfrac{123}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{4049}{36}\right)\sqrt{5}-\dfrac{71}{9}\right)-\left(\left(-\dfrac{5507}{84}\right)\sqrt{5}-9\right)\\
&=&\left(-\dfrac{4049}{36}\right)\sqrt{5}-\dfrac{71}{9}+\left(\dfrac{5507}{84}\right)\sqrt{5}+9\\
&=&\left(-\dfrac{5911}{126}\right)\sqrt{5}+\dfrac{10}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{17}{5}\right)\sqrt{125}+\dfrac{51}{8}+\left(-\dfrac{5}{3}\right)\sqrt{45}+\left(-\dfrac{15}{8}\right)\sqrt{25}+\left(-\dfrac{16}{9}\right)\sqrt{25}+\left(-\dfrac{79}{4}\right)\sqrt{125}+\left(\dfrac{26}{9}\right)\sqrt{20}+\left(\dfrac{17}{4}\right)\sqrt{20}+\left(\dfrac{17}{2}\right)\sqrt{20}+\left(-\dfrac{23}{2}\right)\sqrt{20}+4\right)\times\left(\left(\left(\dfrac{31}{6}\right)\sqrt{125}\right)-\left(\left(\dfrac{18}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{39}{8}\right)\sqrt{20}\right)-\left(\left(5\right)\sqrt{45}\right)-\left(\left(\dfrac{9}{5}\right)\sqrt{25}\right)+\left(-\dfrac{41}{2}\right)\sqrt{45}\right)\\
&=&\left(\left(-17\right)\sqrt{5}+\dfrac{51}{8}+\left(-5\right)\sqrt{5}-\dfrac{75}{8}-\dfrac{80}{9}+\left(-\dfrac{395}{4}\right)\sqrt{5}+\left(\dfrac{52}{9}\right)\sqrt{5}+\left(\dfrac{17}{2}\right)\sqrt{5}+\left(17\right)\sqrt{5}+\left(-23\right)\sqrt{5}+4\right)\times\left(\left(\left(\dfrac{155}{6}\right)\sqrt{5}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{39}{4}\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)-9+\left(-\dfrac{123}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{4049}{36}\right)\sqrt{5}-\dfrac{71}{9}\right)\left(\left(-\dfrac{5507}{84}\right)\sqrt{5}-9\right)\\
&=&\left(\dfrac{22297843}{3024}\right)\sqrt{25}+\left(\dfrac{578129}{378}\right)\sqrt{5}+71\\
&=&\dfrac{111703919}{3024}+\left(\dfrac{578129}{378}\right)\sqrt{5}\\
\end{eqnarray*}