L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(7\right)\sqrt{125}-\dfrac{17}{2}\right)-\left(\left(-\dfrac{11}{6}\right)\sqrt{45}-\dfrac{69}{2}+\left(\dfrac{35}{4}\right)\sqrt{25}\right)\) et \( Y=\left(\dfrac{4}{3}\right)\sqrt{25}+\left(\left(-\dfrac{19}{9}\right)\sqrt{125}\right)-\left(\left(-4\right)\sqrt{45}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{45}\right)\) . 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(7\right)\sqrt{125}-\dfrac{17}{2}\right)-\left(\left(-\dfrac{11}{6}\right)\sqrt{45}-\dfrac{69}{2}+\left(\dfrac{35}{4}\right)\sqrt{25}\right)\right)+\left(\left(\dfrac{4}{3}\right)\sqrt{25}+\left(\left(-\dfrac{19}{9}\right)\sqrt{125}\right)-\left(\left(-4\right)\sqrt{45}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(\left(35\right)\sqrt{5}-\dfrac{17}{2}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{5}-\dfrac{69}{2}+\dfrac{175}{4}\right)\right)+\left(\dfrac{20}{3}+\left(\left(-\dfrac{95}{9}\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(35\right)\sqrt{5}-\dfrac{17}{2}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{5}-\dfrac{69}{2}+\dfrac{175}{4}\right)+\dfrac{20}{3}+\left(\left(-\dfrac{95}{9}\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{148}{9}\right)\sqrt{5}-\dfrac{133}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(7\right)\sqrt{125}-\dfrac{17}{2}\right)-\left(\left(-\dfrac{11}{6}\right)\sqrt{45}-\dfrac{69}{2}+\left(\dfrac{35}{4}\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{25}+\left(\left(-\dfrac{19}{9}\right)\sqrt{125}\right)-\left(\left(-4\right)\sqrt{45}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(\left(35\right)\sqrt{5}-\dfrac{17}{2}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{5}-\dfrac{69}{2}+\dfrac{175}{4}\right)\right)-\left(\dfrac{20}{3}+\left(\left(-\dfrac{95}{9}\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{81}{2}\right)\sqrt{5}-\dfrac{71}{4}\right)-\left(\dfrac{20}{3}+\left(-\dfrac{433}{18}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{81}{2}\right)\sqrt{5}-\dfrac{71}{4}+-\dfrac{20}{3}+\left(\dfrac{433}{18}\right)\sqrt{5}\\
&=&\left(\dfrac{581}{9}\right)\sqrt{5}-\dfrac{293}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(7\right)\sqrt{125}-\dfrac{17}{2}\right)-\left(\left(-\dfrac{11}{6}\right)\sqrt{45}-\dfrac{69}{2}+\left(\dfrac{35}{4}\right)\sqrt{25}\right)\right)\times\left(\left(\dfrac{4}{3}\right)\sqrt{25}+\left(\left(-\dfrac{19}{9}\right)\sqrt{125}\right)-\left(\left(-4\right)\sqrt{45}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(\left(35\right)\sqrt{5}-\dfrac{17}{2}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{5}-\dfrac{69}{2}+\dfrac{175}{4}\right)\right)\times\left(\dfrac{20}{3}+\left(\left(-\dfrac{95}{9}\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{81}{2}\right)\sqrt{5}-\dfrac{71}{4}\right)\left(\dfrac{20}{3}+\left(-\dfrac{433}{18}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{50183}{72}\right)\sqrt{5}+\left(-\dfrac{3897}{4}\right)\sqrt{25}-\dfrac{355}{3}\\
&=&\left(\dfrac{50183}{72}\right)\sqrt{5}-\dfrac{59875}{12}\\
\end{eqnarray*}