L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=-\dfrac{7}{5}+\left(-1\right)\sqrt{28}-\dfrac{25}{6}+\left(\left(-\dfrac{29}{9}\right)\sqrt{28}\right)-\left(\left(-1\right)\sqrt{28}\right)\) et \( Y=\left(\left(\dfrac{3}{5}\right)\sqrt{175}-\dfrac{34}{9}+\left(2\right)\sqrt{28}+\dfrac{65}{4}\right)-\left(\left(\left(-3\right)\sqrt{63}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{49}\right)\right)-\left(\left(\dfrac{59}{3}\right)\sqrt{49}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{7}{5}+\left(-1\right)\sqrt{28}-\dfrac{25}{6}+\left(\left(-\dfrac{29}{9}\right)\sqrt{28}\right)-\left(\left(-1\right)\sqrt{28}\right)\right)+\left(\left(\left(\dfrac{3}{5}\right)\sqrt{175}-\dfrac{34}{9}+\left(2\right)\sqrt{28}+\dfrac{65}{4}\right)-\left(\left(\left(-3\right)\sqrt{63}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{49}\right)\right)-\left(\left(\dfrac{59}{3}\right)\sqrt{49}\right)\right)\\
&=&\left(-\dfrac{7}{5}+\left(-2\right)\sqrt{7}-\dfrac{25}{6}+\left(\left(-\dfrac{58}{9}\right)\sqrt{7}\right)-\left(\left(-2\right)\sqrt{7}\right)\right)+\left(\left(\left(3\right)\sqrt{7}-\dfrac{34}{9}+\left(4\right)\sqrt{7}+\dfrac{65}{4}\right)-\left(\left(\left(-9\right)\sqrt{7}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{7}\right)-\left(\left(-24\right)\sqrt{7}\right)-\dfrac{175}{2}\right)-\dfrac{413}{3}\right)\\
&=&-\dfrac{7}{5}+\left(-2\right)\sqrt{7}-\dfrac{25}{6}+\left(\left(-\dfrac{58}{9}\right)\sqrt{7}\right)-\left(\left(-2\right)\sqrt{7}\right)+\left(\left(3\right)\sqrt{7}-\dfrac{34}{9}+\left(4\right)\sqrt{7}+\dfrac{65}{4}\right)-\left(\left(\left(-9\right)\sqrt{7}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{7}\right)-\left(\left(-24\right)\sqrt{7}\right)-\dfrac{175}{2}\right)-\dfrac{413}{3}\\
&=&-\dfrac{7787}{180}+\left(-\dfrac{215}{18}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{7}{5}+\left(-1\right)\sqrt{28}-\dfrac{25}{6}+\left(\left(-\dfrac{29}{9}\right)\sqrt{28}\right)-\left(\left(-1\right)\sqrt{28}\right)\right)-\left(\left(\left(\dfrac{3}{5}\right)\sqrt{175}-\dfrac{34}{9}+\left(2\right)\sqrt{28}+\dfrac{65}{4}\right)-\left(\left(\left(-3\right)\sqrt{63}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{49}\right)\right)-\left(\left(\dfrac{59}{3}\right)\sqrt{49}\right)\right)\\
&=&\left(-\dfrac{7}{5}+\left(-2\right)\sqrt{7}-\dfrac{25}{6}+\left(\left(-\dfrac{58}{9}\right)\sqrt{7}\right)-\left(\left(-2\right)\sqrt{7}\right)\right)-\left(\left(\left(3\right)\sqrt{7}-\dfrac{34}{9}+\left(4\right)\sqrt{7}+\dfrac{65}{4}\right)-\left(\left(\left(-9\right)\sqrt{7}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{7}\right)-\left(\left(-24\right)\sqrt{7}\right)-\dfrac{175}{2}\right)-\dfrac{413}{3}\right)\\
&=&\left(-\dfrac{167}{30}+\left(-\dfrac{58}{9}\right)\sqrt{7}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{7}-\dfrac{1357}{36}\right)\\
&=&-\dfrac{167}{30}+\left(-\dfrac{58}{9}\right)\sqrt{7}+\left(\dfrac{11}{2}\right)\sqrt{7}+\dfrac{1357}{36}\\
&=&\dfrac{5783}{180}+\left(-\dfrac{17}{18}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{7}{5}+\left(-1\right)\sqrt{28}-\dfrac{25}{6}+\left(\left(-\dfrac{29}{9}\right)\sqrt{28}\right)-\left(\left(-1\right)\sqrt{28}\right)\right)\times\left(\left(\left(\dfrac{3}{5}\right)\sqrt{175}-\dfrac{34}{9}+\left(2\right)\sqrt{28}+\dfrac{65}{4}\right)-\left(\left(\left(-3\right)\sqrt{63}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{49}\right)\right)-\left(\left(\dfrac{59}{3}\right)\sqrt{49}\right)\right)\\
&=&\left(-\dfrac{7}{5}+\left(-2\right)\sqrt{7}-\dfrac{25}{6}+\left(\left(-\dfrac{58}{9}\right)\sqrt{7}\right)-\left(\left(-2\right)\sqrt{7}\right)\right)\times\left(\left(\left(3\right)\sqrt{7}-\dfrac{34}{9}+\left(4\right)\sqrt{7}+\dfrac{65}{4}\right)-\left(\left(\left(-9\right)\sqrt{7}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{7}\right)-\left(\left(-24\right)\sqrt{7}\right)-\dfrac{175}{2}\right)-\dfrac{413}{3}\right)\\
&=&\left(-\dfrac{167}{30}+\left(-\dfrac{58}{9}\right)\sqrt{7}\right)\left(\left(-\dfrac{11}{2}\right)\sqrt{7}-\dfrac{1357}{36}\right)\\
&=&\left(\dfrac{443129}{1620}\right)\sqrt{7}+\dfrac{226619}{1080}+\left(\dfrac{319}{9}\right)\sqrt{49}\\
&=&\left(\dfrac{443129}{1620}\right)\sqrt{7}+\dfrac{494579}{1080}\\
\end{eqnarray*}