L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{39}{4}-\dfrac{46}{7}-\left(\left(\dfrac{47}{6}\right)\sqrt{25}\right)-\left(\left(\dfrac{7}{9}\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{29}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{125}\right)\) et \( Y=\left(-1\right)\sqrt{125}+\left(\dfrac{52}{5}\right)\sqrt{25}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{39}{4}-\dfrac{46}{7}-\left(\left(\dfrac{47}{6}\right)\sqrt{25}\right)-\left(\left(\dfrac{7}{9}\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{29}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{125}\right)\right)+\left(\left(-1\right)\sqrt{125}+\left(\dfrac{52}{5}\right)\sqrt{25}\right)\\
&=&\left(\left(\dfrac{39}{4}-\dfrac{46}{7}-\dfrac{235}{6}-\dfrac{35}{9}\right)-\left(\left(\dfrac{87}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{375}{2}\right)\sqrt{5}\right)\right)+\left(\left(-5\right)\sqrt{5}+52\right)\\
&=&\left(\dfrac{39}{4}-\dfrac{46}{7}-\dfrac{235}{6}-\dfrac{35}{9}\right)-\left(\left(\dfrac{87}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{375}{2}\right)\sqrt{5}\right)+\left(-5\right)\sqrt{5}+52\\
&=&\dfrac{3055}{252}+\left(\dfrac{2381}{14}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{39}{4}-\dfrac{46}{7}-\left(\left(\dfrac{47}{6}\right)\sqrt{25}\right)-\left(\left(\dfrac{7}{9}\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{29}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{125}\right)\right)-\left(\left(-1\right)\sqrt{125}+\left(\dfrac{52}{5}\right)\sqrt{25}\right)\\
&=&\left(\left(\dfrac{39}{4}-\dfrac{46}{7}-\dfrac{235}{6}-\dfrac{35}{9}\right)-\left(\left(\dfrac{87}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{375}{2}\right)\sqrt{5}\right)\right)-\left(\left(-5\right)\sqrt{5}+52\right)\\
&=&\left(-\dfrac{10049}{252}+\left(\dfrac{2451}{14}\right)\sqrt{5}\right)-\left(\left(-5\right)\sqrt{5}+52\right)\\
&=&-\dfrac{10049}{252}+\left(\dfrac{2451}{14}\right)\sqrt{5}+\left(5\right)\sqrt{5}-52\\
&=&-\dfrac{23153}{252}+\left(\dfrac{2521}{14}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{39}{4}-\dfrac{46}{7}-\left(\left(\dfrac{47}{6}\right)\sqrt{25}\right)-\left(\left(\dfrac{7}{9}\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{29}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{125}\right)\right)\times\left(\left(-1\right)\sqrt{125}+\left(\dfrac{52}{5}\right)\sqrt{25}\right)\\
&=&\left(\left(\dfrac{39}{4}-\dfrac{46}{7}-\dfrac{235}{6}-\dfrac{35}{9}\right)-\left(\left(\dfrac{87}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{375}{2}\right)\sqrt{5}\right)\right)\times\left(\left(-5\right)\sqrt{5}+52\right)\\
&=&\left(-\dfrac{10049}{252}+\left(\dfrac{2451}{14}\right)\sqrt{5}\right)\left(\left(-5\right)\sqrt{5}+52\right)\\
&=&\left(\dfrac{2344381}{252}\right)\sqrt{5}-\dfrac{130637}{63}+\left(-\dfrac{12255}{14}\right)\sqrt{25}\\
&=&\left(\dfrac{2344381}{252}\right)\sqrt{5}-\dfrac{116107}{18}\\
\end{eqnarray*}