L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{31}{4}\right)\sqrt{20}-\dfrac{79}{2}-\left(\left(-9\right)\sqrt{20}\right)-\dfrac{72}{7}-\left(\left(-\dfrac{25}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{20}\right)+\left(-\dfrac{58}{7}\right)\sqrt{125}+\left(9\right)\sqrt{45}+3+\left(0\right)\sqrt{125}+\left(-7\right)\sqrt{25}+\left(-\dfrac{40}{9}\right)\sqrt{125}+\left(-\dfrac{20}{9}\right)\sqrt{25}\) et \( Y=\dfrac{71}{5}\) . 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{31}{4}\right)\sqrt{20}-\dfrac{79}{2}-\left(\left(-9\right)\sqrt{20}\right)-\dfrac{72}{7}-\left(\left(-\dfrac{25}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{20}\right)+\left(-\dfrac{58}{7}\right)\sqrt{125}+\left(9\right)\sqrt{45}+3+\left(0\right)\sqrt{125}+\left(-7\right)\sqrt{25}+\left(-\dfrac{40}{9}\right)\sqrt{125}+\left(-\dfrac{20}{9}\right)\sqrt{25}\right)+\left(\dfrac{71}{5}\right)\\
&=&\left(\left(-\dfrac{31}{2}\right)\sqrt{5}-\dfrac{79}{2}-\left(\left(-18\right)\sqrt{5}\right)-\dfrac{72}{7}+\dfrac{125}{4}-\left(\left(-27\right)\sqrt{5}\right)+\left(-\dfrac{290}{7}\right)\sqrt{5}+\left(27\right)\sqrt{5}+3+\left(0\right)\sqrt{5}-35+\left(-\dfrac{200}{9}\right)\sqrt{5}-\dfrac{100}{9}\right)+\left(\dfrac{71}{5}\right)\\
&=&\left(-\dfrac{31}{2}\right)\sqrt{5}-\dfrac{79}{2}-\left(\left(-18\right)\sqrt{5}\right)-\dfrac{72}{7}+\dfrac{125}{4}-\left(\left(-27\right)\sqrt{5}\right)+\left(-\dfrac{290}{7}\right)\sqrt{5}+\left(27\right)\sqrt{5}+3+\left(0\right)\sqrt{5}-35+\left(-\dfrac{200}{9}\right)\sqrt{5}-\dfrac{100}{9}+\dfrac{71}{5}\\
&=&\left(-\dfrac{901}{126}\right)\sqrt{5}-\dfrac{59783}{1260}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{31}{4}\right)\sqrt{20}-\dfrac{79}{2}-\left(\left(-9\right)\sqrt{20}\right)-\dfrac{72}{7}-\left(\left(-\dfrac{25}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{20}\right)+\left(-\dfrac{58}{7}\right)\sqrt{125}+\left(9\right)\sqrt{45}+3+\left(0\right)\sqrt{125}+\left(-7\right)\sqrt{25}+\left(-\dfrac{40}{9}\right)\sqrt{125}+\left(-\dfrac{20}{9}\right)\sqrt{25}\right)-\left(\dfrac{71}{5}\right)\\
&=&\left(\left(-\dfrac{31}{2}\right)\sqrt{5}-\dfrac{79}{2}-\left(\left(-18\right)\sqrt{5}\right)-\dfrac{72}{7}+\dfrac{125}{4}-\left(\left(-27\right)\sqrt{5}\right)+\left(-\dfrac{290}{7}\right)\sqrt{5}+\left(27\right)\sqrt{5}+3+\left(0\right)\sqrt{5}-35+\left(-\dfrac{200}{9}\right)\sqrt{5}-\dfrac{100}{9}\right)-\left(\dfrac{71}{5}\right)\\
&=&\left(\left(-\dfrac{901}{126}\right)\sqrt{5}-\dfrac{15535}{252}\right)-\left(\dfrac{71}{5}\right)\\
&=&\left(-\dfrac{901}{126}\right)\sqrt{5}-\dfrac{15535}{252}+-\dfrac{71}{5}\\
&=&\left(-\dfrac{901}{126}\right)\sqrt{5}-\dfrac{95567}{1260}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{31}{4}\right)\sqrt{20}-\dfrac{79}{2}-\left(\left(-9\right)\sqrt{20}\right)-\dfrac{72}{7}-\left(\left(-\dfrac{25}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{20}\right)+\left(-\dfrac{58}{7}\right)\sqrt{125}+\left(9\right)\sqrt{45}+3+\left(0\right)\sqrt{125}+\left(-7\right)\sqrt{25}+\left(-\dfrac{40}{9}\right)\sqrt{125}+\left(-\dfrac{20}{9}\right)\sqrt{25}\right)\times\left(\dfrac{71}{5}\right)\\
&=&\left(\left(-\dfrac{31}{2}\right)\sqrt{5}-\dfrac{79}{2}-\left(\left(-18\right)\sqrt{5}\right)-\dfrac{72}{7}+\dfrac{125}{4}-\left(\left(-27\right)\sqrt{5}\right)+\left(-\dfrac{290}{7}\right)\sqrt{5}+\left(27\right)\sqrt{5}+3+\left(0\right)\sqrt{5}-35+\left(-\dfrac{200}{9}\right)\sqrt{5}-\dfrac{100}{9}\right)\times\left(\dfrac{71}{5}\right)\\
&=&\left(\left(-\dfrac{901}{126}\right)\sqrt{5}-\dfrac{15535}{252}\right)\left(\dfrac{71}{5}\right)\\
&=&\left(-\dfrac{63971}{630}\right)\sqrt{5}-\dfrac{220597}{252}\\
&=&\left(-\dfrac{63971}{630}\right)\sqrt{5}-\dfrac{220597}{252}\\
\end{eqnarray*}