L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{13}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{45}\right)-\left(\left(4\right)\sqrt{125}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{59}{5}\right)\sqrt{25}+\left(-\dfrac{71}{3}\right)\sqrt{45}+\left(-4\right)\sqrt{125}\right)\) et \( Y=\left(\left(-\dfrac{4}{3}\right)\sqrt{25}+\left(-9\right)\sqrt{20}+\left(-\dfrac{53}{7}\right)\sqrt{25}+\left(\dfrac{59}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{20}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{71}{8}\right)\sqrt{25}\right)\right)\) . 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(\left(\dfrac{13}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{45}\right)-\left(\left(4\right)\sqrt{125}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{59}{5}\right)\sqrt{25}+\left(-\dfrac{71}{3}\right)\sqrt{45}+\left(-4\right)\sqrt{125}\right)\right)+\left(\left(\left(-\dfrac{4}{3}\right)\sqrt{25}+\left(-9\right)\sqrt{20}+\left(-\dfrac{53}{7}\right)\sqrt{25}+\left(\dfrac{59}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{20}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{71}{8}\right)\sqrt{25}\right)\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{65}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{111}{2}\right)\sqrt{5}\right)-\left(\left(20\right)\sqrt{5}\right)-\dfrac{85}{2}\right)-\left(\left(\dfrac{35}{3}\right)\sqrt{5}\right)-\left(-59+\left(-71\right)\sqrt{5}+\left(-20\right)\sqrt{5}\right)\right)+\left(\left(-\dfrac{20}{3}+\left(-18\right)\sqrt{5}-\dfrac{265}{7}+\left(\dfrac{295}{4}\right)\sqrt{5}\right)-\left(\left(-1\right)\sqrt{5}\right)-\left(\dfrac{3}{2}+\dfrac{355}{8}\right)\right)\\
&=&\left(\left(\left(\dfrac{65}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{111}{2}\right)\sqrt{5}\right)-\left(\left(20\right)\sqrt{5}\right)-\dfrac{85}{2}\right)-\left(\left(\dfrac{35}{3}\right)\sqrt{5}\right)-\left(-59+\left(-71\right)\sqrt{5}+\left(-20\right)\sqrt{5}\right)+\left(-\dfrac{20}{3}+\left(-18\right)\sqrt{5}-\dfrac{265}{7}+\left(\dfrac{295}{4}\right)\sqrt{5}\right)-\left(\left(-1\right)\sqrt{5}\right)-\left(\dfrac{3}{2}+\dfrac{355}{8}\right)\\
&=&\left(\dfrac{15193}{84}\right)\sqrt{5}-\dfrac{12415}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{13}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{45}\right)-\left(\left(4\right)\sqrt{125}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{59}{5}\right)\sqrt{25}+\left(-\dfrac{71}{3}\right)\sqrt{45}+\left(-4\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{4}{3}\right)\sqrt{25}+\left(-9\right)\sqrt{20}+\left(-\dfrac{53}{7}\right)\sqrt{25}+\left(\dfrac{59}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{20}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{71}{8}\right)\sqrt{25}\right)\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{65}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{111}{2}\right)\sqrt{5}\right)-\left(\left(20\right)\sqrt{5}\right)-\dfrac{85}{2}\right)-\left(\left(\dfrac{35}{3}\right)\sqrt{5}\right)-\left(-59+\left(-71\right)\sqrt{5}+\left(-20\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{20}{3}+\left(-18\right)\sqrt{5}-\dfrac{265}{7}+\left(\dfrac{295}{4}\right)\sqrt{5}\right)-\left(\left(-1\right)\sqrt{5}\right)-\left(\dfrac{3}{2}+\dfrac{355}{8}\right)\right)\\
&=&\left(\left(\dfrac{5213}{42}\right)\sqrt{5}+\dfrac{33}{2}\right)-\left(-\dfrac{15187}{168}+\left(\dfrac{227}{4}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{5213}{42}\right)\sqrt{5}+\dfrac{33}{2}+\dfrac{15187}{168}+\left(-\dfrac{227}{4}\right)\sqrt{5}\\
&=&\left(\dfrac{5659}{84}\right)\sqrt{5}+\dfrac{17959}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{13}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{45}\right)-\left(\left(4\right)\sqrt{125}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{59}{5}\right)\sqrt{25}+\left(-\dfrac{71}{3}\right)\sqrt{45}+\left(-4\right)\sqrt{125}\right)\right)\times\left(\left(\left(-\dfrac{4}{3}\right)\sqrt{25}+\left(-9\right)\sqrt{20}+\left(-\dfrac{53}{7}\right)\sqrt{25}+\left(\dfrac{59}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{20}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{71}{8}\right)\sqrt{25}\right)\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{65}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{111}{2}\right)\sqrt{5}\right)-\left(\left(20\right)\sqrt{5}\right)-\dfrac{85}{2}\right)-\left(\left(\dfrac{35}{3}\right)\sqrt{5}\right)-\left(-59+\left(-71\right)\sqrt{5}+\left(-20\right)\sqrt{5}\right)\right)\times\left(\left(-\dfrac{20}{3}+\left(-18\right)\sqrt{5}-\dfrac{265}{7}+\left(\dfrac{295}{4}\right)\sqrt{5}\right)-\left(\left(-1\right)\sqrt{5}\right)-\left(\dfrac{3}{2}+\dfrac{355}{8}\right)\right)\\
&=&\left(\left(\dfrac{5213}{42}\right)\sqrt{5}+\dfrac{33}{2}\right)\left(-\dfrac{15187}{168}+\left(\dfrac{227}{4}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{72562769}{7056}\right)\sqrt{5}+\left(\dfrac{1183351}{168}\right)\sqrt{25}-\dfrac{167057}{112}\\
&=&\left(-\dfrac{72562769}{7056}\right)\sqrt{5}+\dfrac{11332339}{336}\\
\end{eqnarray*}