L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{28}{3}\right)\sqrt{125}+\left(\left(-\dfrac{18}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{20}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{62}{9}\right)\sqrt{45}\right)\) et \( Y=\left(\left(-\dfrac{45}{8}\right)\sqrt{25}\right)-\left(\left(6\right)\sqrt{25}\right)+\left(\dfrac{53}{6}\right)\sqrt{45}-\dfrac{9}{2}+\left(8\right)\sqrt{20}+\left(\dfrac{24}{7}\right)\sqrt{20}+\dfrac{52}{5}+\left(\dfrac{24}{7}\right)\sqrt{20}+\left(\dfrac{53}{6}\right)\sqrt{45}+\left(\dfrac{11}{4}\right)\sqrt{125}+\left(-\dfrac{29}{5}\right)\sqrt{125}\) . 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{28}{3}\right)\sqrt{125}+\left(\left(-\dfrac{18}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{20}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{62}{9}\right)\sqrt{45}\right)\right)+\left(\left(\left(-\dfrac{45}{8}\right)\sqrt{25}\right)-\left(\left(6\right)\sqrt{25}\right)+\left(\dfrac{53}{6}\right)\sqrt{45}-\dfrac{9}{2}+\left(8\right)\sqrt{20}+\left(\dfrac{24}{7}\right)\sqrt{20}+\dfrac{52}{5}+\left(\dfrac{24}{7}\right)\sqrt{20}+\left(\dfrac{53}{6}\right)\sqrt{45}+\left(\dfrac{11}{4}\right)\sqrt{125}+\left(-\dfrac{29}{5}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{140}{3}\right)\sqrt{5}+\left(\left(-\dfrac{36}{5}\right)\sqrt{5}\right)-\left(\left(-11\right)\sqrt{5}\right)-\left(\left(-37\right)\sqrt{5}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{62}{3}\right)\sqrt{5}\right)\right)+\left(-\dfrac{225}{8}-30+\left(\dfrac{53}{2}\right)\sqrt{5}-\dfrac{9}{2}+\left(16\right)\sqrt{5}+\left(\dfrac{48}{7}\right)\sqrt{5}+\dfrac{52}{5}+\left(\dfrac{48}{7}\right)\sqrt{5}+\left(\dfrac{53}{2}\right)\sqrt{5}+\left(\dfrac{55}{4}\right)\sqrt{5}+\left(-29\right)\sqrt{5}\right)\\
&=&\left(\dfrac{140}{3}\right)\sqrt{5}+\left(\left(-\dfrac{36}{5}\right)\sqrt{5}\right)-\left(\left(-11\right)\sqrt{5}\right)-\left(\left(-37\right)\sqrt{5}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{62}{3}\right)\sqrt{5}\right)-\dfrac{225}{8}-30+\left(\dfrac{53}{2}\right)\sqrt{5}-\dfrac{9}{2}+\left(16\right)\sqrt{5}+\left(\dfrac{48}{7}\right)\sqrt{5}+\dfrac{52}{5}+\left(\dfrac{48}{7}\right)\sqrt{5}+\left(\dfrac{53}{2}\right)\sqrt{5}+\left(\dfrac{55}{4}\right)\sqrt{5}+\left(-29\right)\sqrt{5}\\
&=&\left(\dfrac{21107}{140}\right)\sqrt{5}-\dfrac{2089}{40}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{28}{3}\right)\sqrt{125}+\left(\left(-\dfrac{18}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{20}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{62}{9}\right)\sqrt{45}\right)\right)-\left(\left(\left(-\dfrac{45}{8}\right)\sqrt{25}\right)-\left(\left(6\right)\sqrt{25}\right)+\left(\dfrac{53}{6}\right)\sqrt{45}-\dfrac{9}{2}+\left(8\right)\sqrt{20}+\left(\dfrac{24}{7}\right)\sqrt{20}+\dfrac{52}{5}+\left(\dfrac{24}{7}\right)\sqrt{20}+\left(\dfrac{53}{6}\right)\sqrt{45}+\left(\dfrac{11}{4}\right)\sqrt{125}+\left(-\dfrac{29}{5}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{140}{3}\right)\sqrt{5}+\left(\left(-\dfrac{36}{5}\right)\sqrt{5}\right)-\left(\left(-11\right)\sqrt{5}\right)-\left(\left(-37\right)\sqrt{5}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{62}{3}\right)\sqrt{5}\right)\right)-\left(-\dfrac{225}{8}-30+\left(\dfrac{53}{2}\right)\sqrt{5}-\dfrac{9}{2}+\left(16\right)\sqrt{5}+\left(\dfrac{48}{7}\right)\sqrt{5}+\dfrac{52}{5}+\left(\dfrac{48}{7}\right)\sqrt{5}+\left(\dfrac{53}{2}\right)\sqrt{5}+\left(\dfrac{55}{4}\right)\sqrt{5}+\left(-29\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{833}{10}\right)\sqrt{5}\right)-\left(-\dfrac{2089}{40}+\left(\dfrac{1889}{28}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{833}{10}\right)\sqrt{5}+\dfrac{2089}{40}+\left(-\dfrac{1889}{28}\right)\sqrt{5}\\
&=&\left(\dfrac{2217}{140}\right)\sqrt{5}+\dfrac{2089}{40}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{28}{3}\right)\sqrt{125}+\left(\left(-\dfrac{18}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{20}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{62}{9}\right)\sqrt{45}\right)\right)\times\left(\left(\left(-\dfrac{45}{8}\right)\sqrt{25}\right)-\left(\left(6\right)\sqrt{25}\right)+\left(\dfrac{53}{6}\right)\sqrt{45}-\dfrac{9}{2}+\left(8\right)\sqrt{20}+\left(\dfrac{24}{7}\right)\sqrt{20}+\dfrac{52}{5}+\left(\dfrac{24}{7}\right)\sqrt{20}+\left(\dfrac{53}{6}\right)\sqrt{45}+\left(\dfrac{11}{4}\right)\sqrt{125}+\left(-\dfrac{29}{5}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{140}{3}\right)\sqrt{5}+\left(\left(-\dfrac{36}{5}\right)\sqrt{5}\right)-\left(\left(-11\right)\sqrt{5}\right)-\left(\left(-37\right)\sqrt{5}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{62}{3}\right)\sqrt{5}\right)\right)\times\left(-\dfrac{225}{8}-30+\left(\dfrac{53}{2}\right)\sqrt{5}-\dfrac{9}{2}+\left(16\right)\sqrt{5}+\left(\dfrac{48}{7}\right)\sqrt{5}+\dfrac{52}{5}+\left(\dfrac{48}{7}\right)\sqrt{5}+\left(\dfrac{53}{2}\right)\sqrt{5}+\left(\dfrac{55}{4}\right)\sqrt{5}+\left(-29\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{833}{10}\right)\sqrt{5}\right)\left(-\dfrac{2089}{40}+\left(\dfrac{1889}{28}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{1740137}{400}\right)\sqrt{5}+\left(\dfrac{224791}{40}\right)\sqrt{25}\\
&=&\left(-\dfrac{1740137}{400}\right)\sqrt{5}+\dfrac{224791}{8}\\
\end{eqnarray*}