L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\dfrac{21}{2}\right)\sqrt{45}+\left(9\right)\sqrt{125}+\left(0\right)\sqrt{45}+\left(\dfrac{17}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{30}{7}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{26}{3}\right)\sqrt{125}\right)-\left(\left(-6\right)\sqrt{20}\right)-\left(\left(-\dfrac{24}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{37}{5}\right)\sqrt{20}\right)\right)\) et \( Y=\dfrac{10}{7}-\left(\left(\left(\dfrac{41}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{19}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{37}{6}\right)\sqrt{20}\right)\right)-\left(\left(\left(\dfrac{53}{7}\right)\sqrt{20}\right)+6\right)\) . 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{21}{2}\right)\sqrt{45}+\left(9\right)\sqrt{125}+\left(0\right)\sqrt{45}+\left(\dfrac{17}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{30}{7}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{26}{3}\right)\sqrt{125}\right)-\left(\left(-6\right)\sqrt{20}\right)-\left(\left(-\dfrac{24}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{37}{5}\right)\sqrt{20}\right)\right)\right)+\left(\dfrac{10}{7}-\left(\left(\left(\dfrac{41}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{19}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{37}{6}\right)\sqrt{20}\right)\right)-\left(\left(\left(\dfrac{53}{7}\right)\sqrt{20}\right)+6\right)\right)\\
&=&\left(\left(\left(\dfrac{63}{2}\right)\sqrt{5}+\left(45\right)\sqrt{5}+\left(0\right)\sqrt{5}+\dfrac{85}{2}\right)-\left(\left(-\dfrac{150}{7}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{130}{3}\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{74}{5}\right)\sqrt{5}\right)\right)\right)+\left(\dfrac{10}{7}-\left(\dfrac{205}{2}-\left(\left(-\dfrac{19}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{5}\right)\right)-\left(\left(\left(\dfrac{106}{7}\right)\sqrt{5}\right)+6\right)\right)\\
&=&\left(\left(\dfrac{63}{2}\right)\sqrt{5}+\left(45\right)\sqrt{5}+\left(0\right)\sqrt{5}+\dfrac{85}{2}\right)-\left(\left(-\dfrac{150}{7}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{130}{3}\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{74}{5}\right)\sqrt{5}\right)\right)+\dfrac{10}{7}-\left(\dfrac{205}{2}-\left(\left(-\dfrac{19}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{5}\right)\right)-\left(\left(\left(\dfrac{106}{7}\right)\sqrt{5}\right)+6\right)\\
&=&\left(\dfrac{484}{5}\right)\sqrt{5}-\dfrac{452}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{21}{2}\right)\sqrt{45}+\left(9\right)\sqrt{125}+\left(0\right)\sqrt{45}+\left(\dfrac{17}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{30}{7}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{26}{3}\right)\sqrt{125}\right)-\left(\left(-6\right)\sqrt{20}\right)-\left(\left(-\dfrac{24}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{37}{5}\right)\sqrt{20}\right)\right)\right)-\left(\dfrac{10}{7}-\left(\left(\left(\dfrac{41}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{19}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{37}{6}\right)\sqrt{20}\right)\right)-\left(\left(\left(\dfrac{53}{7}\right)\sqrt{20}\right)+6\right)\right)\\
&=&\left(\left(\left(\dfrac{63}{2}\right)\sqrt{5}+\left(45\right)\sqrt{5}+\left(0\right)\sqrt{5}+\dfrac{85}{2}\right)-\left(\left(-\dfrac{150}{7}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{130}{3}\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{74}{5}\right)\sqrt{5}\right)\right)\right)-\left(\dfrac{10}{7}-\left(\dfrac{205}{2}-\left(\left(-\dfrac{19}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{5}\right)\right)-\left(\left(\left(\dfrac{106}{7}\right)\sqrt{5}\right)+6\right)\right)\\
&=&\left(\left(\dfrac{28093}{210}\right)\sqrt{5}+\dfrac{85}{2}\right)-\left(-\dfrac{1499}{14}+\left(-\dfrac{1553}{42}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{28093}{210}\right)\sqrt{5}+\dfrac{85}{2}+\dfrac{1499}{14}+\left(\dfrac{1553}{42}\right)\sqrt{5}\\
&=&\left(\dfrac{17929}{105}\right)\sqrt{5}+\dfrac{1047}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{21}{2}\right)\sqrt{45}+\left(9\right)\sqrt{125}+\left(0\right)\sqrt{45}+\left(\dfrac{17}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{30}{7}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{26}{3}\right)\sqrt{125}\right)-\left(\left(-6\right)\sqrt{20}\right)-\left(\left(-\dfrac{24}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{37}{5}\right)\sqrt{20}\right)\right)\right)\times\left(\dfrac{10}{7}-\left(\left(\left(\dfrac{41}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{19}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{37}{6}\right)\sqrt{20}\right)\right)-\left(\left(\left(\dfrac{53}{7}\right)\sqrt{20}\right)+6\right)\right)\\
&=&\left(\left(\left(\dfrac{63}{2}\right)\sqrt{5}+\left(45\right)\sqrt{5}+\left(0\right)\sqrt{5}+\dfrac{85}{2}\right)-\left(\left(-\dfrac{150}{7}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{130}{3}\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{74}{5}\right)\sqrt{5}\right)\right)\right)\times\left(\dfrac{10}{7}-\left(\dfrac{205}{2}-\left(\left(-\dfrac{19}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{5}\right)\right)-\left(\left(\left(\dfrac{106}{7}\right)\sqrt{5}\right)+6\right)\right)\\
&=&\left(\left(\dfrac{28093}{210}\right)\sqrt{5}+\dfrac{85}{2}\right)\left(-\dfrac{1499}{14}+\left(-\dfrac{1553}{42}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{3925452888}{246960}\right)\sqrt{5}+\left(-\dfrac{43628429}{8820}\right)\sqrt{25}-\dfrac{127415}{28}\\
&=&\left(-\dfrac{3925452888}{246960}\right)\sqrt{5}-\dfrac{25827787}{882}\\
\end{eqnarray*}