L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{69}{4}\right)\sqrt{20}\right)+\dfrac{62}{5}-\left(\left(\dfrac{49}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{49}{3}\right)\sqrt{45}\right)\right)-\left(\left(\left(3\right)\sqrt{125}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{20}\right)-\dfrac{21}{2}-\left(\left(-\dfrac{73}{4}\right)\sqrt{45}\right)\right)\) et \( Y=\left(\dfrac{26}{7}\right)\sqrt{25}\) . 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{69}{4}\right)\sqrt{20}\right)+\dfrac{62}{5}-\left(\left(\dfrac{49}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{49}{3}\right)\sqrt{45}\right)\right)-\left(\left(\left(3\right)\sqrt{125}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{20}\right)-\dfrac{21}{2}-\left(\left(-\dfrac{73}{4}\right)\sqrt{45}\right)\right)\right)+\left(\left(\dfrac{26}{7}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\left(\dfrac{69}{2}\right)\sqrt{5}\right)+\dfrac{62}{5}-\left(\left(49\right)\sqrt{5}\right)-\left(\left(49\right)\sqrt{5}\right)\right)-\left(\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)-\dfrac{21}{2}-\left(\left(-\dfrac{219}{4}\right)\sqrt{5}\right)\right)\right)+\left(\dfrac{130}{7}\right)\\
&=&\left(\left(\left(\dfrac{69}{2}\right)\sqrt{5}\right)+\dfrac{62}{5}-\left(\left(49\right)\sqrt{5}\right)-\left(\left(49\right)\sqrt{5}\right)\right)-\left(\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)-\dfrac{21}{2}-\left(\left(-\dfrac{219}{4}\right)\sqrt{5}\right)\right)+\dfrac{130}{7}\\
&=&\left(-\dfrac{579}{4}\right)\sqrt{5}+\dfrac{2903}{70}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{69}{4}\right)\sqrt{20}\right)+\dfrac{62}{5}-\left(\left(\dfrac{49}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{49}{3}\right)\sqrt{45}\right)\right)-\left(\left(\left(3\right)\sqrt{125}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{20}\right)-\dfrac{21}{2}-\left(\left(-\dfrac{73}{4}\right)\sqrt{45}\right)\right)\right)-\left(\left(\dfrac{26}{7}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\left(\dfrac{69}{2}\right)\sqrt{5}\right)+\dfrac{62}{5}-\left(\left(49\right)\sqrt{5}\right)-\left(\left(49\right)\sqrt{5}\right)\right)-\left(\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)-\dfrac{21}{2}-\left(\left(-\dfrac{219}{4}\right)\sqrt{5}\right)\right)\right)-\left(\dfrac{130}{7}\right)\\
&=&\left(\left(-\dfrac{579}{4}\right)\sqrt{5}+\dfrac{229}{10}\right)-\left(\dfrac{130}{7}\right)\\
&=&\left(-\dfrac{579}{4}\right)\sqrt{5}+\dfrac{229}{10}+-\dfrac{130}{7}\\
&=&\left(-\dfrac{579}{4}\right)\sqrt{5}+\dfrac{303}{70}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{69}{4}\right)\sqrt{20}\right)+\dfrac{62}{5}-\left(\left(\dfrac{49}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{49}{3}\right)\sqrt{45}\right)\right)-\left(\left(\left(3\right)\sqrt{125}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{20}\right)-\dfrac{21}{2}-\left(\left(-\dfrac{73}{4}\right)\sqrt{45}\right)\right)\right)\times\left(\left(\dfrac{26}{7}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\left(\dfrac{69}{2}\right)\sqrt{5}\right)+\dfrac{62}{5}-\left(\left(49\right)\sqrt{5}\right)-\left(\left(49\right)\sqrt{5}\right)\right)-\left(\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)-\dfrac{21}{2}-\left(\left(-\dfrac{219}{4}\right)\sqrt{5}\right)\right)\right)\times\left(\dfrac{130}{7}\right)\\
&=&\left(\left(-\dfrac{579}{4}\right)\sqrt{5}+\dfrac{229}{10}\right)\left(\dfrac{130}{7}\right)\\
&=&\left(-\dfrac{37635}{14}\right)\sqrt{5}+\dfrac{2977}{7}\\
&=&\left(-\dfrac{37635}{14}\right)\sqrt{5}+\dfrac{2977}{7}\\
\end{eqnarray*}