L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{17}{9}\) et \( Y=\left(\left(7\right)\sqrt{20}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{20}\right)-\left(\left(-\dfrac{19}{6}\right)\sqrt{45}\right)-\left(\left(\dfrac{17}{5}\right)\sqrt{20}\right)+\dfrac{23}{2}+\left(\left(4\right)\sqrt{20}\right)-\left(\left(\dfrac{79}{8}\right)\sqrt{25}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{17}{9}\right)+\left(\left(\left(7\right)\sqrt{20}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{20}\right)-\left(\left(-\dfrac{19}{6}\right)\sqrt{45}\right)-\left(\left(\dfrac{17}{5}\right)\sqrt{20}\right)+\dfrac{23}{2}+\left(\left(4\right)\sqrt{20}\right)-\left(\left(\dfrac{79}{8}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{17}{9}\right)+\left(\left(\left(14\right)\sqrt{5}\right)-\left(\left(-\dfrac{275}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{106}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{5}\right)+\dfrac{23}{2}+\left(\left(8\right)\sqrt{5}\right)-\dfrac{395}{8}\right)\\
&=&\dfrac{17}{9}+\left(\left(14\right)\sqrt{5}\right)-\left(\left(-\dfrac{275}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{106}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{5}\right)+\dfrac{23}{2}+\left(\left(8\right)\sqrt{5}\right)-\dfrac{395}{8}\\
&=&-\dfrac{2591}{72}+\left(\dfrac{15203}{140}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{17}{9}\right)-\left(\left(\left(7\right)\sqrt{20}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{20}\right)-\left(\left(-\dfrac{19}{6}\right)\sqrt{45}\right)-\left(\left(\dfrac{17}{5}\right)\sqrt{20}\right)+\dfrac{23}{2}+\left(\left(4\right)\sqrt{20}\right)-\left(\left(\dfrac{79}{8}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{17}{9}\right)-\left(\left(\left(14\right)\sqrt{5}\right)-\left(\left(-\dfrac{275}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{106}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{5}\right)+\dfrac{23}{2}+\left(\left(8\right)\sqrt{5}\right)-\dfrac{395}{8}\right)\\
&=&\left(\dfrac{17}{9}\right)-\left(\left(\dfrac{15203}{140}\right)\sqrt{5}-\dfrac{303}{8}\right)\\
&=&\dfrac{17}{9}+\left(-\dfrac{15203}{140}\right)\sqrt{5}+\dfrac{303}{8}\\
&=&\dfrac{2863}{72}+\left(-\dfrac{15203}{140}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{17}{9}\right)\times\left(\left(\left(7\right)\sqrt{20}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{20}\right)-\left(\left(-\dfrac{19}{6}\right)\sqrt{45}\right)-\left(\left(\dfrac{17}{5}\right)\sqrt{20}\right)+\dfrac{23}{2}+\left(\left(4\right)\sqrt{20}\right)-\left(\left(\dfrac{79}{8}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{17}{9}\right)\times\left(\left(\left(14\right)\sqrt{5}\right)-\left(\left(-\dfrac{275}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{106}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{5}\right)+\dfrac{23}{2}+\left(\left(8\right)\sqrt{5}\right)-\dfrac{395}{8}\right)\\
&=&\left(\dfrac{17}{9}\right)\left(\left(\dfrac{15203}{140}\right)\sqrt{5}-\dfrac{303}{8}\right)\\
&=&\left(\dfrac{258451}{1260}\right)\sqrt{5}-\dfrac{1717}{24}\\
&=&\left(\dfrac{258451}{1260}\right)\sqrt{5}-\dfrac{1717}{24}\\
\end{eqnarray*}