L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{43}{7}\right)\sqrt{125}\) et \( Y=\left(\left(-\dfrac{50}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{12}{5}\right)\sqrt{45}+\left(\dfrac{19}{2}\right)\sqrt{20}+\left(-\dfrac{1}{3}\right)\sqrt{125}+\left(-\dfrac{81}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{63}{4}\right)\sqrt{20}\right)\) . 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{43}{7}\right)\sqrt{125}\right)+\left(\left(\left(-\dfrac{50}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{12}{5}\right)\sqrt{45}+\left(\dfrac{19}{2}\right)\sqrt{20}+\left(-\dfrac{1}{3}\right)\sqrt{125}+\left(-\dfrac{81}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{63}{4}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(-\dfrac{215}{7}\right)\sqrt{5}\right)+\left(-\dfrac{250}{3}-\left(\left(-\dfrac{36}{5}\right)\sqrt{5}+\left(19\right)\sqrt{5}+\left(-\dfrac{5}{3}\right)\sqrt{5}+\left(-\dfrac{243}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{63}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{215}{7}\right)\sqrt{5}-\dfrac{250}{3}-\left(\left(-\dfrac{36}{5}\right)\sqrt{5}+\left(19\right)\sqrt{5}+\left(-\dfrac{5}{3}\right)\sqrt{5}+\left(-\dfrac{243}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{63}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{2447}{210}\right)\sqrt{5}-\dfrac{250}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{43}{7}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{50}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{12}{5}\right)\sqrt{45}+\left(\dfrac{19}{2}\right)\sqrt{20}+\left(-\dfrac{1}{3}\right)\sqrt{125}+\left(-\dfrac{81}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{63}{4}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(-\dfrac{215}{7}\right)\sqrt{5}\right)-\left(-\dfrac{250}{3}-\left(\left(-\dfrac{36}{5}\right)\sqrt{5}+\left(19\right)\sqrt{5}+\left(-\dfrac{5}{3}\right)\sqrt{5}+\left(-\dfrac{243}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{63}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{215}{7}\right)\sqrt{5}\right)-\left(-\dfrac{250}{3}+\left(\dfrac{1271}{30}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{215}{7}\right)\sqrt{5}+\dfrac{250}{3}+\left(-\dfrac{1271}{30}\right)\sqrt{5}\\
&=&\left(-\dfrac{15347}{210}\right)\sqrt{5}+\dfrac{250}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{43}{7}\right)\sqrt{125}\right)\times\left(\left(\left(-\dfrac{50}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{12}{5}\right)\sqrt{45}+\left(\dfrac{19}{2}\right)\sqrt{20}+\left(-\dfrac{1}{3}\right)\sqrt{125}+\left(-\dfrac{81}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{63}{4}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(-\dfrac{215}{7}\right)\sqrt{5}\right)\times\left(-\dfrac{250}{3}-\left(\left(-\dfrac{36}{5}\right)\sqrt{5}+\left(19\right)\sqrt{5}+\left(-\dfrac{5}{3}\right)\sqrt{5}+\left(-\dfrac{243}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{63}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{215}{7}\right)\sqrt{5}\right)\left(-\dfrac{250}{3}+\left(\dfrac{1271}{30}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{53750}{21}\right)\sqrt{5}+\left(-\dfrac{54653}{42}\right)\sqrt{25}\\
&=&\left(\dfrac{53750}{21}\right)\sqrt{5}-\dfrac{273265}{42}\\
\end{eqnarray*}