L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(1\right)\sqrt{25}+\left(\dfrac{17}{2}\right)\sqrt{25}+\left(8\right)\sqrt{45}+\left(-\dfrac{31}{9}\right)\sqrt{125}\right)-\left(-\dfrac{3}{7}+\left(-9\right)\sqrt{45}-\dfrac{25}{8}+\left(\dfrac{43}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{31}{9}\right)\sqrt{125}\right)\) et \( Y=\left(-3\right)\sqrt{25}\) . 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(1\right)\sqrt{25}+\left(\dfrac{17}{2}\right)\sqrt{25}+\left(8\right)\sqrt{45}+\left(-\dfrac{31}{9}\right)\sqrt{125}\right)-\left(-\dfrac{3}{7}+\left(-9\right)\sqrt{45}-\dfrac{25}{8}+\left(\dfrac{43}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{31}{9}\right)\sqrt{125}\right)\right)+\left(\left(-3\right)\sqrt{25}\right)\\
&=&\left(\left(5+\dfrac{85}{2}+\left(24\right)\sqrt{5}+\left(-\dfrac{155}{9}\right)\sqrt{5}\right)-\left(-\dfrac{3}{7}+\left(-27\right)\sqrt{5}-\dfrac{25}{8}+\left(\dfrac{215}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{155}{9}\right)\sqrt{5}\right)\right)+\left(-15\right)\\
&=&\left(5+\dfrac{85}{2}+\left(24\right)\sqrt{5}+\left(-\dfrac{155}{9}\right)\sqrt{5}\right)-\left(-\dfrac{3}{7}+\left(-27\right)\sqrt{5}-\dfrac{25}{8}+\left(\dfrac{215}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{155}{9}\right)\sqrt{5}\right)-15\\
&=&\dfrac{2019}{56}+\left(\dfrac{142}{7}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(1\right)\sqrt{25}+\left(\dfrac{17}{2}\right)\sqrt{25}+\left(8\right)\sqrt{45}+\left(-\dfrac{31}{9}\right)\sqrt{125}\right)-\left(-\dfrac{3}{7}+\left(-9\right)\sqrt{45}-\dfrac{25}{8}+\left(\dfrac{43}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{31}{9}\right)\sqrt{125}\right)\right)-\left(\left(-3\right)\sqrt{25}\right)\\
&=&\left(\left(5+\dfrac{85}{2}+\left(24\right)\sqrt{5}+\left(-\dfrac{155}{9}\right)\sqrt{5}\right)-\left(-\dfrac{3}{7}+\left(-27\right)\sqrt{5}-\dfrac{25}{8}+\left(\dfrac{215}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{155}{9}\right)\sqrt{5}\right)\right)-\left(-15\right)\\
&=&\left(\dfrac{2859}{56}+\left(\dfrac{142}{7}\right)\sqrt{5}\right)-\left(-15\right)\\
&=&\dfrac{2859}{56}+\left(\dfrac{142}{7}\right)\sqrt{5}+15\\
&=&\dfrac{3699}{56}+\left(\dfrac{142}{7}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(1\right)\sqrt{25}+\left(\dfrac{17}{2}\right)\sqrt{25}+\left(8\right)\sqrt{45}+\left(-\dfrac{31}{9}\right)\sqrt{125}\right)-\left(-\dfrac{3}{7}+\left(-9\right)\sqrt{45}-\dfrac{25}{8}+\left(\dfrac{43}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{31}{9}\right)\sqrt{125}\right)\right)\times\left(\left(-3\right)\sqrt{25}\right)\\
&=&\left(\left(5+\dfrac{85}{2}+\left(24\right)\sqrt{5}+\left(-\dfrac{155}{9}\right)\sqrt{5}\right)-\left(-\dfrac{3}{7}+\left(-27\right)\sqrt{5}-\dfrac{25}{8}+\left(\dfrac{215}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{155}{9}\right)\sqrt{5}\right)\right)\times\left(-15\right)\\
&=&\left(\dfrac{2859}{56}+\left(\dfrac{142}{7}\right)\sqrt{5}\right)\left(-15\right)\\
&=&-\dfrac{42885}{56}+\left(-\dfrac{2130}{7}\right)\sqrt{5}\\
&=&-\dfrac{42885}{56}+\left(-\dfrac{2130}{7}\right)\sqrt{5}\\
\end{eqnarray*}