L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{3}{7}\right)\sqrt{125}\) et \( Y=\left(\left(7\right)\sqrt{20}\right)-\left(\left(\dfrac{36}{5}\right)\sqrt{45}\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{45}+\left(\dfrac{1}{8}\right)\sqrt{45}\right)-\left(\left(\dfrac{1}{9}\right)\sqrt{20}\right)-\left(\dfrac{63}{8}+\left(-\dfrac{7}{6}\right)\sqrt{20}+\left(-\dfrac{57}{7}\right)\sqrt{20}-\dfrac{39}{4}\right)\) . 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{3}{7}\right)\sqrt{125}\right)+\left(\left(\left(7\right)\sqrt{20}\right)-\left(\left(\dfrac{36}{5}\right)\sqrt{45}\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{45}+\left(\dfrac{1}{8}\right)\sqrt{45}\right)-\left(\left(\dfrac{1}{9}\right)\sqrt{20}\right)-\left(\dfrac{63}{8}+\left(-\dfrac{7}{6}\right)\sqrt{20}+\left(-\dfrac{57}{7}\right)\sqrt{20}-\dfrac{39}{4}\right)\right)\\
&=&\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)+\left(\left(\left(14\right)\sqrt{5}\right)-\left(\left(\dfrac{108}{5}\right)\sqrt{5}\right)-\left(\left(26\right)\sqrt{5}+\left(\dfrac{3}{8}\right)\sqrt{5}\right)-\left(\left(\dfrac{2}{9}\right)\sqrt{5}\right)-\left(\dfrac{63}{8}+\left(-\dfrac{7}{3}\right)\sqrt{5}+\left(-\dfrac{114}{7}\right)\sqrt{5}-\dfrac{39}{4}\right)\right)\\
&=&\left(-\dfrac{15}{7}\right)\sqrt{5}+\left(\left(14\right)\sqrt{5}\right)-\left(\left(\dfrac{108}{5}\right)\sqrt{5}\right)-\left(\left(26\right)\sqrt{5}+\left(\dfrac{3}{8}\right)\sqrt{5}\right)-\left(\left(\dfrac{2}{9}\right)\sqrt{5}\right)-\left(\dfrac{63}{8}+\left(-\dfrac{7}{3}\right)\sqrt{5}+\left(-\dfrac{114}{7}\right)\sqrt{5}-\dfrac{39}{4}\right)\\
&=&\left(-\dfrac{44657}{2520}\right)\sqrt{5}+\dfrac{15}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{3}{7}\right)\sqrt{125}\right)-\left(\left(\left(7\right)\sqrt{20}\right)-\left(\left(\dfrac{36}{5}\right)\sqrt{45}\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{45}+\left(\dfrac{1}{8}\right)\sqrt{45}\right)-\left(\left(\dfrac{1}{9}\right)\sqrt{20}\right)-\left(\dfrac{63}{8}+\left(-\dfrac{7}{6}\right)\sqrt{20}+\left(-\dfrac{57}{7}\right)\sqrt{20}-\dfrac{39}{4}\right)\right)\\
&=&\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)-\left(\left(\left(14\right)\sqrt{5}\right)-\left(\left(\dfrac{108}{5}\right)\sqrt{5}\right)-\left(\left(26\right)\sqrt{5}+\left(\dfrac{3}{8}\right)\sqrt{5}\right)-\left(\left(\dfrac{2}{9}\right)\sqrt{5}\right)-\left(\dfrac{63}{8}+\left(-\dfrac{7}{3}\right)\sqrt{5}+\left(-\dfrac{114}{7}\right)\sqrt{5}-\dfrac{39}{4}\right)\right)\\
&=&\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{39257}{2520}\right)\sqrt{5}+\dfrac{15}{8}\right)\\
&=&\left(-\dfrac{15}{7}\right)\sqrt{5}+\left(\dfrac{39257}{2520}\right)\sqrt{5}-\dfrac{15}{8}\\
&=&\left(\dfrac{33857}{2520}\right)\sqrt{5}-\dfrac{15}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{3}{7}\right)\sqrt{125}\right)\times\left(\left(\left(7\right)\sqrt{20}\right)-\left(\left(\dfrac{36}{5}\right)\sqrt{45}\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{45}+\left(\dfrac{1}{8}\right)\sqrt{45}\right)-\left(\left(\dfrac{1}{9}\right)\sqrt{20}\right)-\left(\dfrac{63}{8}+\left(-\dfrac{7}{6}\right)\sqrt{20}+\left(-\dfrac{57}{7}\right)\sqrt{20}-\dfrac{39}{4}\right)\right)\\
&=&\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)\times\left(\left(\left(14\right)\sqrt{5}\right)-\left(\left(\dfrac{108}{5}\right)\sqrt{5}\right)-\left(\left(26\right)\sqrt{5}+\left(\dfrac{3}{8}\right)\sqrt{5}\right)-\left(\left(\dfrac{2}{9}\right)\sqrt{5}\right)-\left(\dfrac{63}{8}+\left(-\dfrac{7}{3}\right)\sqrt{5}+\left(-\dfrac{114}{7}\right)\sqrt{5}-\dfrac{39}{4}\right)\right)\\
&=&\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)\left(\left(-\dfrac{39257}{2520}\right)\sqrt{5}+\dfrac{15}{8}\right)\\
&=&\left(\dfrac{39257}{1176}\right)\sqrt{25}+\left(-\dfrac{225}{56}\right)\sqrt{5}\\
&=&\dfrac{196285}{1176}+\left(-\dfrac{225}{56}\right)\sqrt{5}\\
\end{eqnarray*}