L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(-3\right)\sqrt{45}\) et \( Y=\left(\left(\dfrac{15}{2}\right)\sqrt{25}+\left(\dfrac{65}{2}\right)\sqrt{45}-9-9+\left(\dfrac{65}{2}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{64}{3}\right)\sqrt{45}\right)-0\right)-\left(-9-\left(\left(-8\right)\sqrt{125}\right)\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(\dfrac{71}{3}\right)\sqrt{125}+\left(\dfrac{5}{6}\right)\sqrt{45}+\left(-\dfrac{80}{7}\right)\sqrt{125}+\left(\dfrac{67}{4}\right)\sqrt{45}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-3\right)\sqrt{45}\right)+\left(\left(\left(\dfrac{15}{2}\right)\sqrt{25}+\left(\dfrac{65}{2}\right)\sqrt{45}-9-9+\left(\dfrac{65}{2}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{64}{3}\right)\sqrt{45}\right)-0\right)-\left(-9-\left(\left(-8\right)\sqrt{125}\right)\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(\dfrac{71}{3}\right)\sqrt{125}+\left(\dfrac{5}{6}\right)\sqrt{45}+\left(-\dfrac{80}{7}\right)\sqrt{125}+\left(\dfrac{67}{4}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(-9\right)\sqrt{5}\right)+\left(\left(\dfrac{75}{2}+\left(\dfrac{195}{2}\right)\sqrt{5}-9-9+\left(\dfrac{195}{2}\right)\sqrt{5}\right)-\left(\left(\left(-64\right)\sqrt{5}\right)-0\right)-\left(-9-\left(\left(-40\right)\sqrt{5}\right)\right)-\left(-\dfrac{115}{3}+\left(\dfrac{355}{3}\right)\sqrt{5}+\left(\dfrac{5}{2}\right)\sqrt{5}+\left(-\dfrac{400}{7}\right)\sqrt{5}+\left(\dfrac{201}{4}\right)\sqrt{5}\right)\right)\\
&=&\left(-9\right)\sqrt{5}+\left(\dfrac{75}{2}+\left(\dfrac{195}{2}\right)\sqrt{5}-9-9+\left(\dfrac{195}{2}\right)\sqrt{5}\right)-\left(\left(\left(-64\right)\sqrt{5}\right)-0\right)-\left(-9-\left(\left(-40\right)\sqrt{5}\right)\right)-\left(-\dfrac{115}{3}+\left(\dfrac{355}{3}\right)\sqrt{5}+\left(\dfrac{5}{2}\right)\sqrt{5}+\left(-\dfrac{400}{7}\right)\sqrt{5}+\left(\dfrac{201}{4}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{8069}{84}\right)\sqrt{5}+\dfrac{401}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-3\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{15}{2}\right)\sqrt{25}+\left(\dfrac{65}{2}\right)\sqrt{45}-9-9+\left(\dfrac{65}{2}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{64}{3}\right)\sqrt{45}\right)-0\right)-\left(-9-\left(\left(-8\right)\sqrt{125}\right)\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(\dfrac{71}{3}\right)\sqrt{125}+\left(\dfrac{5}{6}\right)\sqrt{45}+\left(-\dfrac{80}{7}\right)\sqrt{125}+\left(\dfrac{67}{4}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(-9\right)\sqrt{5}\right)-\left(\left(\dfrac{75}{2}+\left(\dfrac{195}{2}\right)\sqrt{5}-9-9+\left(\dfrac{195}{2}\right)\sqrt{5}\right)-\left(\left(\left(-64\right)\sqrt{5}\right)-0\right)-\left(-9-\left(\left(-40\right)\sqrt{5}\right)\right)-\left(-\dfrac{115}{3}+\left(\dfrac{355}{3}\right)\sqrt{5}+\left(\dfrac{5}{2}\right)\sqrt{5}+\left(-\dfrac{400}{7}\right)\sqrt{5}+\left(\dfrac{201}{4}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-9\right)\sqrt{5}\right)-\left(\dfrac{401}{6}+\left(\dfrac{8825}{84}\right)\sqrt{5}\right)\\
&=&\left(-9\right)\sqrt{5}+-\dfrac{401}{6}+\left(-\dfrac{8825}{84}\right)\sqrt{5}\\
&=&\left(-\dfrac{9581}{84}\right)\sqrt{5}-\dfrac{401}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-3\right)\sqrt{45}\right)\times\left(\left(\left(\dfrac{15}{2}\right)\sqrt{25}+\left(\dfrac{65}{2}\right)\sqrt{45}-9-9+\left(\dfrac{65}{2}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{64}{3}\right)\sqrt{45}\right)-0\right)-\left(-9-\left(\left(-8\right)\sqrt{125}\right)\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(\dfrac{71}{3}\right)\sqrt{125}+\left(\dfrac{5}{6}\right)\sqrt{45}+\left(-\dfrac{80}{7}\right)\sqrt{125}+\left(\dfrac{67}{4}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(-9\right)\sqrt{5}\right)\times\left(\left(\dfrac{75}{2}+\left(\dfrac{195}{2}\right)\sqrt{5}-9-9+\left(\dfrac{195}{2}\right)\sqrt{5}\right)-\left(\left(\left(-64\right)\sqrt{5}\right)-0\right)-\left(-9-\left(\left(-40\right)\sqrt{5}\right)\right)-\left(-\dfrac{115}{3}+\left(\dfrac{355}{3}\right)\sqrt{5}+\left(\dfrac{5}{2}\right)\sqrt{5}+\left(-\dfrac{400}{7}\right)\sqrt{5}+\left(\dfrac{201}{4}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-9\right)\sqrt{5}\right)\left(\dfrac{401}{6}+\left(\dfrac{8825}{84}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{1203}{2}\right)\sqrt{5}+\left(-\dfrac{26475}{28}\right)\sqrt{25}\\
&=&\left(-\dfrac{1203}{2}\right)\sqrt{5}-\dfrac{132375}{28}\\
\end{eqnarray*}