L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(8\right)\sqrt{25}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{61}{4}\right)\sqrt{20}\right)\right)-\left(\dfrac{29}{5}-4\right)\) et \( Y=\left(\dfrac{79}{4}\right)\sqrt{125}+\left(\dfrac{39}{2}\right)\sqrt{20}+\left(-3\right)\sqrt{20}+\left(\dfrac{20}{7}\right)\sqrt{45}+\left(\left(-\dfrac{47}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{67}{8}\right)\sqrt{45}\right)\) . 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(8\right)\sqrt{25}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{61}{4}\right)\sqrt{20}\right)\right)-\left(\dfrac{29}{5}-4\right)\right)+\left(\left(\dfrac{79}{4}\right)\sqrt{125}+\left(\dfrac{39}{2}\right)\sqrt{20}+\left(-3\right)\sqrt{20}+\left(\dfrac{20}{7}\right)\sqrt{45}+\left(\left(-\dfrac{47}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{67}{8}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(40-\left(\left(\dfrac{145}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{61}{2}\right)\sqrt{5}\right)\right)-\left(\dfrac{29}{5}-4\right)\right)+\left(\left(\dfrac{395}{4}\right)\sqrt{5}+\left(39\right)\sqrt{5}+\left(-6\right)\sqrt{5}+\left(\dfrac{60}{7}\right)\sqrt{5}-\dfrac{235}{4}-\left(\left(\dfrac{201}{8}\right)\sqrt{5}\right)\right)\\
&=&\left(40-\left(\left(\dfrac{145}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{61}{2}\right)\sqrt{5}\right)\right)-\left(\dfrac{29}{5}-4\right)+\left(\dfrac{395}{4}\right)\sqrt{5}+\left(39\right)\sqrt{5}+\left(-6\right)\sqrt{5}+\left(\dfrac{60}{7}\right)\sqrt{5}-\dfrac{235}{4}-\left(\left(\dfrac{201}{8}\right)\sqrt{5}\right)\\
&=&-\dfrac{411}{20}+\left(\dfrac{2713}{56}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(8\right)\sqrt{25}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{61}{4}\right)\sqrt{20}\right)\right)-\left(\dfrac{29}{5}-4\right)\right)-\left(\left(\dfrac{79}{4}\right)\sqrt{125}+\left(\dfrac{39}{2}\right)\sqrt{20}+\left(-3\right)\sqrt{20}+\left(\dfrac{20}{7}\right)\sqrt{45}+\left(\left(-\dfrac{47}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{67}{8}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(40-\left(\left(\dfrac{145}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{61}{2}\right)\sqrt{5}\right)\right)-\left(\dfrac{29}{5}-4\right)\right)-\left(\left(\dfrac{395}{4}\right)\sqrt{5}+\left(39\right)\sqrt{5}+\left(-6\right)\sqrt{5}+\left(\dfrac{60}{7}\right)\sqrt{5}-\dfrac{235}{4}-\left(\left(\dfrac{201}{8}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{191}{5}+\left(-\dfrac{267}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{6451}{56}\right)\sqrt{5}-\dfrac{235}{4}\right)\\
&=&\dfrac{191}{5}+\left(-\dfrac{267}{4}\right)\sqrt{5}+\left(-\dfrac{6451}{56}\right)\sqrt{5}+\dfrac{235}{4}\\
&=&\dfrac{1939}{20}+\left(-\dfrac{10189}{56}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(8\right)\sqrt{25}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{61}{4}\right)\sqrt{20}\right)\right)-\left(\dfrac{29}{5}-4\right)\right)\times\left(\left(\dfrac{79}{4}\right)\sqrt{125}+\left(\dfrac{39}{2}\right)\sqrt{20}+\left(-3\right)\sqrt{20}+\left(\dfrac{20}{7}\right)\sqrt{45}+\left(\left(-\dfrac{47}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{67}{8}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(40-\left(\left(\dfrac{145}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{61}{2}\right)\sqrt{5}\right)\right)-\left(\dfrac{29}{5}-4\right)\right)\times\left(\left(\dfrac{395}{4}\right)\sqrt{5}+\left(39\right)\sqrt{5}+\left(-6\right)\sqrt{5}+\left(\dfrac{60}{7}\right)\sqrt{5}-\dfrac{235}{4}-\left(\left(\dfrac{201}{8}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{191}{5}+\left(-\dfrac{267}{4}\right)\sqrt{5}\right)\left(\left(\dfrac{6451}{56}\right)\sqrt{5}-\dfrac{235}{4}\right)\\
&=&\left(\dfrac{4660357}{560}\right)\sqrt{5}-\dfrac{8977}{4}+\left(-\dfrac{1722417}{224}\right)\sqrt{25}\\
&=&\left(\dfrac{4660357}{560}\right)\sqrt{5}-\dfrac{9114797}{224}\\
\end{eqnarray*}