L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(0\right)\sqrt{20}-\dfrac{23}{9}\) et \( Y=\left(\left(-6\right)\sqrt{45}\right)-\left(\left(6\right)\sqrt{45}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{45}\right)+\left(\left(\dfrac{65}{3}\right)\sqrt{45}\right)-\left(\left(9\right)\sqrt{20}\right)-\dfrac{3}{8}+\dfrac{80}{9}+\dfrac{50}{9}+\left(8\right)\sqrt{20}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(0\right)\sqrt{20}-\dfrac{23}{9}\right)+\left(\left(\left(-6\right)\sqrt{45}\right)-\left(\left(6\right)\sqrt{45}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{45}\right)+\left(\left(\dfrac{65}{3}\right)\sqrt{45}\right)-\left(\left(9\right)\sqrt{20}\right)-\dfrac{3}{8}+\dfrac{80}{9}+\dfrac{50}{9}+\left(8\right)\sqrt{20}\right)\\
&=&\left(\left(0\right)\sqrt{5}-\dfrac{23}{9}\right)+\left(\left(\left(-18\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{5}\right)+\left(\left(65\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)-\dfrac{3}{8}+\dfrac{80}{9}+\dfrac{50}{9}+\left(16\right)\sqrt{5}\right)\\
&=&\left(0\right)\sqrt{5}-\dfrac{23}{9}+\left(\left(-18\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{5}\right)+\left(\left(65\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)-\dfrac{3}{8}+\dfrac{80}{9}+\dfrac{50}{9}+\left(16\right)\sqrt{5}\\
&=&\left(\dfrac{1265}{14}\right)\sqrt{5}+\dfrac{829}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(0\right)\sqrt{20}-\dfrac{23}{9}\right)-\left(\left(\left(-6\right)\sqrt{45}\right)-\left(\left(6\right)\sqrt{45}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{45}\right)+\left(\left(\dfrac{65}{3}\right)\sqrt{45}\right)-\left(\left(9\right)\sqrt{20}\right)-\dfrac{3}{8}+\dfrac{80}{9}+\dfrac{50}{9}+\left(8\right)\sqrt{20}\right)\\
&=&\left(\left(0\right)\sqrt{5}-\dfrac{23}{9}\right)-\left(\left(\left(-18\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{5}\right)+\left(\left(65\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)-\dfrac{3}{8}+\dfrac{80}{9}+\dfrac{50}{9}+\left(16\right)\sqrt{5}\right)\\
&=&\left(\left(0\right)\sqrt{5}-\dfrac{23}{9}\right)-\left(\left(\dfrac{1265}{14}\right)\sqrt{5}+\dfrac{1013}{72}\right)\\
&=&\left(0\right)\sqrt{5}-\dfrac{23}{9}+\left(-\dfrac{1265}{14}\right)\sqrt{5}-\dfrac{1013}{72}\\
&=&\left(-\dfrac{1265}{14}\right)\sqrt{5}-\dfrac{133}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(0\right)\sqrt{20}-\dfrac{23}{9}\right)\times\left(\left(\left(-6\right)\sqrt{45}\right)-\left(\left(6\right)\sqrt{45}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{45}\right)+\left(\left(\dfrac{65}{3}\right)\sqrt{45}\right)-\left(\left(9\right)\sqrt{20}\right)-\dfrac{3}{8}+\dfrac{80}{9}+\dfrac{50}{9}+\left(8\right)\sqrt{20}\right)\\
&=&\left(\left(0\right)\sqrt{5}-\dfrac{23}{9}\right)\times\left(\left(\left(-18\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{5}\right)+\left(\left(65\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)-\dfrac{3}{8}+\dfrac{80}{9}+\dfrac{50}{9}+\left(16\right)\sqrt{5}\right)\\
&=&\left(\left(0\right)\sqrt{5}-\dfrac{23}{9}\right)\left(\left(\dfrac{1265}{14}\right)\sqrt{5}+\dfrac{1013}{72}\right)\\
&=&\left(0\right)\sqrt{25}+\left(-\dfrac{29095}{126}\right)\sqrt{5}-\dfrac{23299}{648}\\
&=&-\dfrac{23299}{648}+\left(-\dfrac{29095}{126}\right)\sqrt{5}\\
\end{eqnarray*}