L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-\dfrac{22}{5}\right)\sqrt{20}+\left(-8\right)\sqrt{20}+\left(-8\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{45}-3\right)-\left(-\dfrac{29}{2}+\dfrac{38}{9}-\left(\left(\dfrac{59}{6}\right)\sqrt{45}\right)\right)\) et \( Y=\left(\left(\dfrac{10}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)+\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{53}{5}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{125}\right)\) . 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(-\dfrac{22}{5}\right)\sqrt{20}+\left(-8\right)\sqrt{20}+\left(-8\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{45}-3\right)-\left(-\dfrac{29}{2}+\dfrac{38}{9}-\left(\left(\dfrac{59}{6}\right)\sqrt{45}\right)\right)\right)+\left(\left(\left(\dfrac{10}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)+\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{53}{5}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(-\dfrac{44}{5}\right)\sqrt{5}+\left(-16\right)\sqrt{5}+\left(-16\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}-3\right)-\left(-\dfrac{29}{2}+\dfrac{38}{9}-\left(\left(\dfrac{59}{2}\right)\sqrt{5}\right)\right)\right)+\left(\left(\left(\dfrac{50}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)+\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{159}{5}\right)\sqrt{5}\right)-\left(\left(10\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{44}{5}\right)\sqrt{5}+\left(-16\right)\sqrt{5}+\left(-16\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}-3\right)-\left(-\dfrac{29}{2}+\dfrac{38}{9}-\left(\left(\dfrac{59}{2}\right)\sqrt{5}\right)\right)+\left(\left(\dfrac{50}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)+\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{159}{5}\right)\sqrt{5}\right)-\left(\left(10\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{1633}{30}\right)\sqrt{5}+\dfrac{239}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{22}{5}\right)\sqrt{20}+\left(-8\right)\sqrt{20}+\left(-8\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{45}-3\right)-\left(-\dfrac{29}{2}+\dfrac{38}{9}-\left(\left(\dfrac{59}{6}\right)\sqrt{45}\right)\right)\right)-\left(\left(\left(\dfrac{10}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)+\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{53}{5}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(-\dfrac{44}{5}\right)\sqrt{5}+\left(-16\right)\sqrt{5}+\left(-16\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}-3\right)-\left(-\dfrac{29}{2}+\dfrac{38}{9}-\left(\left(\dfrac{59}{2}\right)\sqrt{5}\right)\right)\right)-\left(\left(\left(\dfrac{50}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)+\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{159}{5}\right)\sqrt{5}\right)-\left(\left(10\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{293}{10}\right)\sqrt{5}+\dfrac{239}{18}\right)-\left(\left(-\dfrac{377}{15}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{293}{10}\right)\sqrt{5}+\dfrac{239}{18}+\left(\dfrac{377}{15}\right)\sqrt{5}\\
&=&\left(-\dfrac{25}{6}\right)\sqrt{5}+\dfrac{239}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{22}{5}\right)\sqrt{20}+\left(-8\right)\sqrt{20}+\left(-8\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{45}-3\right)-\left(-\dfrac{29}{2}+\dfrac{38}{9}-\left(\left(\dfrac{59}{6}\right)\sqrt{45}\right)\right)\right)\times\left(\left(\left(\dfrac{10}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)+\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{53}{5}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(-\dfrac{44}{5}\right)\sqrt{5}+\left(-16\right)\sqrt{5}+\left(-16\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}-3\right)-\left(-\dfrac{29}{2}+\dfrac{38}{9}-\left(\left(\dfrac{59}{2}\right)\sqrt{5}\right)\right)\right)\times\left(\left(\left(\dfrac{50}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)+\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{159}{5}\right)\sqrt{5}\right)-\left(\left(10\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{293}{10}\right)\sqrt{5}+\dfrac{239}{18}\right)\left(\left(-\dfrac{377}{15}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{110461}{150}\right)\sqrt{25}+\left(-\dfrac{90103}{270}\right)\sqrt{5}\\
&=&\dfrac{110461}{30}+\left(-\dfrac{90103}{270}\right)\sqrt{5}\\
\end{eqnarray*}