L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{37}{2}\right)\sqrt{50}+\left(\dfrac{31}{9}\right)\sqrt{4}+\left(\left(\dfrac{19}{5}\right)\sqrt{4}\right)-\dfrac{27}{4}-\left(\left(-1\right)\sqrt{4}\right)-2+\left(\dfrac{31}{9}\right)\sqrt{50}\) et \( Y=\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{50}{7}\right)\sqrt{50}\right)+\left(2\right)\sqrt{4}+\dfrac{25}{3}+\left(-\dfrac{64}{5}\right)\sqrt{50}\) . 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{37}{2}\right)\sqrt{50}+\left(\dfrac{31}{9}\right)\sqrt{4}+\left(\left(\dfrac{19}{5}\right)\sqrt{4}\right)-\dfrac{27}{4}-\left(\left(-1\right)\sqrt{4}\right)-2+\left(\dfrac{31}{9}\right)\sqrt{50}\right)+\left(\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{50}{7}\right)\sqrt{50}\right)+\left(2\right)\sqrt{4}+\dfrac{25}{3}+\left(-\dfrac{64}{5}\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{185}{2}\right)\sqrt{2}+\dfrac{62}{9}+\dfrac{38}{5}-\dfrac{27}{4}+2-2+\left(\dfrac{155}{9}\right)\sqrt{2}\right)+\left(\left(\left(22\right)\sqrt{2}\right)-\left(\left(\dfrac{250}{7}\right)\sqrt{2}\right)+4+\dfrac{25}{3}+\left(-64\right)\sqrt{2}\right)\\
&=&\left(\dfrac{185}{2}\right)\sqrt{2}+\dfrac{62}{9}+\dfrac{38}{5}-\dfrac{27}{4}+2-2+\left(\dfrac{155}{9}\right)\sqrt{2}+\left(\left(22\right)\sqrt{2}\right)-\left(\left(\dfrac{250}{7}\right)\sqrt{2}\right)+4+\dfrac{25}{3}+\left(-64\right)\sqrt{2}\\
&=&\left(\dfrac{4033}{126}\right)\sqrt{2}+\dfrac{3613}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{37}{2}\right)\sqrt{50}+\left(\dfrac{31}{9}\right)\sqrt{4}+\left(\left(\dfrac{19}{5}\right)\sqrt{4}\right)-\dfrac{27}{4}-\left(\left(-1\right)\sqrt{4}\right)-2+\left(\dfrac{31}{9}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{50}{7}\right)\sqrt{50}\right)+\left(2\right)\sqrt{4}+\dfrac{25}{3}+\left(-\dfrac{64}{5}\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{185}{2}\right)\sqrt{2}+\dfrac{62}{9}+\dfrac{38}{5}-\dfrac{27}{4}+2-2+\left(\dfrac{155}{9}\right)\sqrt{2}\right)-\left(\left(\left(22\right)\sqrt{2}\right)-\left(\left(\dfrac{250}{7}\right)\sqrt{2}\right)+4+\dfrac{25}{3}+\left(-64\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{1975}{18}\right)\sqrt{2}+\dfrac{1393}{180}\right)-\left(\left(-\dfrac{544}{7}\right)\sqrt{2}+\dfrac{37}{3}\right)\\
&=&\left(\dfrac{1975}{18}\right)\sqrt{2}+\dfrac{1393}{180}+\left(\dfrac{544}{7}\right)\sqrt{2}-\dfrac{37}{3}\\
&=&\left(\dfrac{23617}{126}\right)\sqrt{2}-\dfrac{827}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{37}{2}\right)\sqrt{50}+\left(\dfrac{31}{9}\right)\sqrt{4}+\left(\left(\dfrac{19}{5}\right)\sqrt{4}\right)-\dfrac{27}{4}-\left(\left(-1\right)\sqrt{4}\right)-2+\left(\dfrac{31}{9}\right)\sqrt{50}\right)\times\left(\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{50}{7}\right)\sqrt{50}\right)+\left(2\right)\sqrt{4}+\dfrac{25}{3}+\left(-\dfrac{64}{5}\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{185}{2}\right)\sqrt{2}+\dfrac{62}{9}+\dfrac{38}{5}-\dfrac{27}{4}+2-2+\left(\dfrac{155}{9}\right)\sqrt{2}\right)\times\left(\left(\left(22\right)\sqrt{2}\right)-\left(\left(\dfrac{250}{7}\right)\sqrt{2}\right)+4+\dfrac{25}{3}+\left(-64\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{1975}{18}\right)\sqrt{2}+\dfrac{1393}{180}\right)\left(\left(-\dfrac{544}{7}\right)\sqrt{2}+\dfrac{37}{3}\right)\\
&=&\left(-\dfrac{537200}{63}\right)\sqrt{4}+\left(\dfrac{202991}{270}\right)\sqrt{2}+\dfrac{51541}{540}\\
&=&-\dfrac{64103213}{3780}+\left(\dfrac{202991}{270}\right)\sqrt{2}\\
\end{eqnarray*}