L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{20}{7}\right)\sqrt{20}+\left(-\dfrac{43}{4}\right)\sqrt{25}+\left(-\dfrac{36}{7}\right)\sqrt{45}+\left(-6\right)\sqrt{25}\right)-\left(\left(9\right)\sqrt{45}+\left(-\dfrac{1}{7}\right)\sqrt{20}+\dfrac{33}{5}+\left(-\dfrac{73}{8}\right)\sqrt{45}\right)\) et \( Y=-\dfrac{47}{5}+\dfrac{33}{7}-\left(\left(-\dfrac{77}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{45}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{20}+\left(-\dfrac{43}{4}\right)\sqrt{25}+\left(-\dfrac{36}{7}\right)\sqrt{45}+\left(-6\right)\sqrt{25}\right)-\left(\left(9\right)\sqrt{45}+\left(-\dfrac{1}{7}\right)\sqrt{20}+\dfrac{33}{5}+\left(-\dfrac{73}{8}\right)\sqrt{45}\right)\right)+\left(-\dfrac{47}{5}+\dfrac{33}{7}-\left(\left(-\dfrac{77}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(\left(-\dfrac{40}{7}\right)\sqrt{5}-\dfrac{215}{4}+\left(-\dfrac{108}{7}\right)\sqrt{5}-30\right)-\left(\left(27\right)\sqrt{5}+\left(-\dfrac{2}{7}\right)\sqrt{5}+\dfrac{33}{5}+\left(-\dfrac{219}{8}\right)\sqrt{5}\right)\right)+\left(-\dfrac{47}{5}+\dfrac{33}{7}+77-\left(\left(11\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{40}{7}\right)\sqrt{5}-\dfrac{215}{4}+\left(-\dfrac{108}{7}\right)\sqrt{5}-30\right)-\left(\left(27\right)\sqrt{5}+\left(-\dfrac{2}{7}\right)\sqrt{5}+\dfrac{33}{5}+\left(-\dfrac{219}{8}\right)\sqrt{5}\right)-\dfrac{47}{5}+\dfrac{33}{7}+77-\left(\left(11\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{1763}{56}\right)\sqrt{5}-\dfrac{505}{28}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{20}+\left(-\dfrac{43}{4}\right)\sqrt{25}+\left(-\dfrac{36}{7}\right)\sqrt{45}+\left(-6\right)\sqrt{25}\right)-\left(\left(9\right)\sqrt{45}+\left(-\dfrac{1}{7}\right)\sqrt{20}+\dfrac{33}{5}+\left(-\dfrac{73}{8}\right)\sqrt{45}\right)\right)-\left(-\dfrac{47}{5}+\dfrac{33}{7}-\left(\left(-\dfrac{77}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(\left(-\dfrac{40}{7}\right)\sqrt{5}-\dfrac{215}{4}+\left(-\dfrac{108}{7}\right)\sqrt{5}-30\right)-\left(\left(27\right)\sqrt{5}+\left(-\dfrac{2}{7}\right)\sqrt{5}+\dfrac{33}{5}+\left(-\dfrac{219}{8}\right)\sqrt{5}\right)\right)-\left(-\dfrac{47}{5}+\dfrac{33}{7}+77-\left(\left(11\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{1147}{56}\right)\sqrt{5}-\dfrac{1807}{20}\right)-\left(\dfrac{2531}{35}+\left(-11\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{1147}{56}\right)\sqrt{5}-\dfrac{1807}{20}+-\dfrac{2531}{35}+\left(11\right)\sqrt{5}\\
&=&\left(-\dfrac{531}{56}\right)\sqrt{5}-\dfrac{22773}{140}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{20}+\left(-\dfrac{43}{4}\right)\sqrt{25}+\left(-\dfrac{36}{7}\right)\sqrt{45}+\left(-6\right)\sqrt{25}\right)-\left(\left(9\right)\sqrt{45}+\left(-\dfrac{1}{7}\right)\sqrt{20}+\dfrac{33}{5}+\left(-\dfrac{73}{8}\right)\sqrt{45}\right)\right)\times\left(-\dfrac{47}{5}+\dfrac{33}{7}-\left(\left(-\dfrac{77}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(\left(-\dfrac{40}{7}\right)\sqrt{5}-\dfrac{215}{4}+\left(-\dfrac{108}{7}\right)\sqrt{5}-30\right)-\left(\left(27\right)\sqrt{5}+\left(-\dfrac{2}{7}\right)\sqrt{5}+\dfrac{33}{5}+\left(-\dfrac{219}{8}\right)\sqrt{5}\right)\right)\times\left(-\dfrac{47}{5}+\dfrac{33}{7}+77-\left(\left(11\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{1147}{56}\right)\sqrt{5}-\dfrac{1807}{20}\right)\left(\dfrac{2531}{35}+\left(-11\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{955111}{1960}\right)\sqrt{5}+\left(\dfrac{12617}{56}\right)\sqrt{25}-\dfrac{4573517}{700}\\
&=&\left(-\dfrac{955111}{1960}\right)\sqrt{5}-\dfrac{7569909}{1400}\\
\end{eqnarray*}