L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(6\right)\sqrt{20}+\left(\dfrac{36}{5}\right)\sqrt{25}+\left(-\dfrac{13}{2}\right)\sqrt{125}+\left(-\dfrac{1}{3}\right)\sqrt{20}+\left(6\right)\sqrt{20}-3+\dfrac{63}{2}-\left(\left(\dfrac{39}{8}\right)\sqrt{45}\right)\) et \( Y=\left(\left(3\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{125}\right)+\left(\left(-\dfrac{76}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{77}{9}\right)\sqrt{45}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(6\right)\sqrt{20}+\left(\dfrac{36}{5}\right)\sqrt{25}+\left(-\dfrac{13}{2}\right)\sqrt{125}+\left(-\dfrac{1}{3}\right)\sqrt{20}+\left(6\right)\sqrt{20}-3+\dfrac{63}{2}-\left(\left(\dfrac{39}{8}\right)\sqrt{45}\right)\right)+\left(\left(\left(3\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{125}\right)+\left(\left(-\dfrac{76}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{77}{9}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(12\right)\sqrt{5}+36+\left(-\dfrac{65}{2}\right)\sqrt{5}+\left(-\dfrac{2}{3}\right)\sqrt{5}+\left(12\right)\sqrt{5}-3+\dfrac{63}{2}-\left(\left(\dfrac{117}{8}\right)\sqrt{5}\right)\right)+\left(\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{5}\right)+\left(\left(-\dfrac{152}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{222}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{77}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(12\right)\sqrt{5}+36+\left(-\dfrac{65}{2}\right)\sqrt{5}+\left(-\dfrac{2}{3}\right)\sqrt{5}+\left(12\right)\sqrt{5}-3+\dfrac{63}{2}-\left(\left(\dfrac{117}{8}\right)\sqrt{5}\right)+\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{5}\right)+\left(\left(-\dfrac{152}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{222}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{77}{3}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{6385}{168}\right)\sqrt{5}+\dfrac{129}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(6\right)\sqrt{20}+\left(\dfrac{36}{5}\right)\sqrt{25}+\left(-\dfrac{13}{2}\right)\sqrt{125}+\left(-\dfrac{1}{3}\right)\sqrt{20}+\left(6\right)\sqrt{20}-3+\dfrac{63}{2}-\left(\left(\dfrac{39}{8}\right)\sqrt{45}\right)\right)-\left(\left(\left(3\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{125}\right)+\left(\left(-\dfrac{76}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{77}{9}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(12\right)\sqrt{5}+36+\left(-\dfrac{65}{2}\right)\sqrt{5}+\left(-\dfrac{2}{3}\right)\sqrt{5}+\left(12\right)\sqrt{5}-3+\dfrac{63}{2}-\left(\left(\dfrac{117}{8}\right)\sqrt{5}\right)\right)-\left(\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{5}\right)+\left(\left(-\dfrac{152}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{222}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{77}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{571}{24}\right)\sqrt{5}+\dfrac{129}{2}\right)-\left(\left(-\dfrac{199}{14}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{571}{24}\right)\sqrt{5}+\dfrac{129}{2}+\left(\dfrac{199}{14}\right)\sqrt{5}\\
&=&\left(-\dfrac{1609}{168}\right)\sqrt{5}+\dfrac{129}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(6\right)\sqrt{20}+\left(\dfrac{36}{5}\right)\sqrt{25}+\left(-\dfrac{13}{2}\right)\sqrt{125}+\left(-\dfrac{1}{3}\right)\sqrt{20}+\left(6\right)\sqrt{20}-3+\dfrac{63}{2}-\left(\left(\dfrac{39}{8}\right)\sqrt{45}\right)\right)\times\left(\left(\left(3\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{125}\right)+\left(\left(-\dfrac{76}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{77}{9}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(12\right)\sqrt{5}+36+\left(-\dfrac{65}{2}\right)\sqrt{5}+\left(-\dfrac{2}{3}\right)\sqrt{5}+\left(12\right)\sqrt{5}-3+\dfrac{63}{2}-\left(\left(\dfrac{117}{8}\right)\sqrt{5}\right)\right)\times\left(\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{5}\right)+\left(\left(-\dfrac{152}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{222}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{77}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{571}{24}\right)\sqrt{5}+\dfrac{129}{2}\right)\left(\left(-\dfrac{199}{14}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{113629}{336}\right)\sqrt{25}+\left(-\dfrac{25671}{28}\right)\sqrt{5}\\
&=&\dfrac{568145}{336}+\left(-\dfrac{25671}{28}\right)\sqrt{5}\\
\end{eqnarray*}