L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{65}{3}\right)\sqrt{45}+\left(-\dfrac{77}{6}\right)\sqrt{20}+\left(\dfrac{29}{4}\right)\sqrt{45}+\left(\dfrac{49}{9}\right)\sqrt{20}-9+\dfrac{6}{5}-\dfrac{29}{8}+\left(\dfrac{11}{2}\right)\sqrt{125}+\left(\dfrac{23}{3}\right)\sqrt{20}\) et \( Y=\left(\left(\left(\dfrac{16}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{20}\right)+\dfrac{77}{2}\right)-\left(-\dfrac{11}{2}-\left(\left(\dfrac{4}{9}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{31}{7}\right)\sqrt{20}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{65}{3}\right)\sqrt{45}+\left(-\dfrac{77}{6}\right)\sqrt{20}+\left(\dfrac{29}{4}\right)\sqrt{45}+\left(\dfrac{49}{9}\right)\sqrt{20}-9+\dfrac{6}{5}-\dfrac{29}{8}+\left(\dfrac{11}{2}\right)\sqrt{125}+\left(\dfrac{23}{3}\right)\sqrt{20}\right)+\left(\left(\left(\left(\dfrac{16}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{20}\right)+\dfrac{77}{2}\right)-\left(-\dfrac{11}{2}-\left(\left(\dfrac{4}{9}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{31}{7}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(-65\right)\sqrt{5}+\left(-\dfrac{77}{3}\right)\sqrt{5}+\left(\dfrac{87}{4}\right)\sqrt{5}+\left(\dfrac{98}{9}\right)\sqrt{5}-9+\dfrac{6}{5}-\dfrac{29}{8}+\left(\dfrac{55}{2}\right)\sqrt{5}+\left(\dfrac{46}{3}\right)\sqrt{5}\right)+\left(\left(\dfrac{80}{3}-\left(\left(-\dfrac{21}{2}\right)\sqrt{5}\right)+\dfrac{77}{2}\right)-\left(-\dfrac{11}{2}-\left(\left(\dfrac{8}{9}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{5}\right)\right)\\
&=&\left(-65\right)\sqrt{5}+\left(-\dfrac{77}{3}\right)\sqrt{5}+\left(\dfrac{87}{4}\right)\sqrt{5}+\left(\dfrac{98}{9}\right)\sqrt{5}-9+\dfrac{6}{5}-\dfrac{29}{8}+\left(\dfrac{55}{2}\right)\sqrt{5}+\left(\dfrac{46}{3}\right)\sqrt{5}+\left(\dfrac{80}{3}-\left(\left(-\dfrac{21}{2}\right)\sqrt{5}\right)+\dfrac{77}{2}\right)-\left(-\dfrac{11}{2}-\left(\left(\dfrac{8}{9}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{3191}{252}\right)\sqrt{5}+\dfrac{7109}{120}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{65}{3}\right)\sqrt{45}+\left(-\dfrac{77}{6}\right)\sqrt{20}+\left(\dfrac{29}{4}\right)\sqrt{45}+\left(\dfrac{49}{9}\right)\sqrt{20}-9+\dfrac{6}{5}-\dfrac{29}{8}+\left(\dfrac{11}{2}\right)\sqrt{125}+\left(\dfrac{23}{3}\right)\sqrt{20}\right)-\left(\left(\left(\left(\dfrac{16}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{20}\right)+\dfrac{77}{2}\right)-\left(-\dfrac{11}{2}-\left(\left(\dfrac{4}{9}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{31}{7}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(-65\right)\sqrt{5}+\left(-\dfrac{77}{3}\right)\sqrt{5}+\left(\dfrac{87}{4}\right)\sqrt{5}+\left(\dfrac{98}{9}\right)\sqrt{5}-9+\dfrac{6}{5}-\dfrac{29}{8}+\left(\dfrac{55}{2}\right)\sqrt{5}+\left(\dfrac{46}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{80}{3}-\left(\left(-\dfrac{21}{2}\right)\sqrt{5}\right)+\dfrac{77}{2}\right)-\left(-\dfrac{11}{2}-\left(\left(\dfrac{8}{9}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{547}{36}\right)\sqrt{5}-\dfrac{457}{40}\right)-\left(\dfrac{212}{3}+\left(\dfrac{319}{126}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{547}{36}\right)\sqrt{5}-\dfrac{457}{40}+-\dfrac{212}{3}+\left(-\dfrac{319}{126}\right)\sqrt{5}\\
&=&\left(-\dfrac{1489}{84}\right)\sqrt{5}-\dfrac{9851}{120}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{65}{3}\right)\sqrt{45}+\left(-\dfrac{77}{6}\right)\sqrt{20}+\left(\dfrac{29}{4}\right)\sqrt{45}+\left(\dfrac{49}{9}\right)\sqrt{20}-9+\dfrac{6}{5}-\dfrac{29}{8}+\left(\dfrac{11}{2}\right)\sqrt{125}+\left(\dfrac{23}{3}\right)\sqrt{20}\right)\times\left(\left(\left(\left(\dfrac{16}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{20}\right)+\dfrac{77}{2}\right)-\left(-\dfrac{11}{2}-\left(\left(\dfrac{4}{9}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{31}{7}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(-65\right)\sqrt{5}+\left(-\dfrac{77}{3}\right)\sqrt{5}+\left(\dfrac{87}{4}\right)\sqrt{5}+\left(\dfrac{98}{9}\right)\sqrt{5}-9+\dfrac{6}{5}-\dfrac{29}{8}+\left(\dfrac{55}{2}\right)\sqrt{5}+\left(\dfrac{46}{3}\right)\sqrt{5}\right)\times\left(\left(\dfrac{80}{3}-\left(\left(-\dfrac{21}{2}\right)\sqrt{5}\right)+\dfrac{77}{2}\right)-\left(-\dfrac{11}{2}-\left(\left(\dfrac{8}{9}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{547}{36}\right)\sqrt{5}-\dfrac{457}{40}\right)\left(\dfrac{212}{3}+\left(\dfrac{319}{126}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{16672309}{15120}\right)\sqrt{5}+\left(-\dfrac{174493}{4536}\right)\sqrt{25}-\dfrac{24221}{30}\\
&=&\left(-\dfrac{16672309}{15120}\right)\sqrt{5}-\dfrac{22673401}{22680}\\
\end{eqnarray*}