L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-\dfrac{65}{3}\right)\sqrt{12}\right)-\dfrac{13}{3}+\dfrac{16}{9}+\dfrac{18}{5}\right)-\left(\left(\left(-2\right)\sqrt{12}\right)-\left(\left(-\dfrac{9}{7}\right)\sqrt{75}\right)\right)\) et \( Y=\left(\dfrac{19}{3}\right)\sqrt{12}\) . 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(\left(-\dfrac{65}{3}\right)\sqrt{12}\right)-\dfrac{13}{3}+\dfrac{16}{9}+\dfrac{18}{5}\right)-\left(\left(\left(-2\right)\sqrt{12}\right)-\left(\left(-\dfrac{9}{7}\right)\sqrt{75}\right)\right)\right)+\left(\left(\dfrac{19}{3}\right)\sqrt{12}\right)\\
&=&\left(\left(\left(\left(-\dfrac{130}{3}\right)\sqrt{3}\right)-\dfrac{13}{3}+\dfrac{16}{9}+\dfrac{18}{5}\right)-\left(\left(\left(-4\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{7}\right)\sqrt{3}\right)\right)\right)+\left(\left(\dfrac{38}{3}\right)\sqrt{3}\right)\\
&=&\left(\left(\left(-\dfrac{130}{3}\right)\sqrt{3}\right)-\dfrac{13}{3}+\dfrac{16}{9}+\dfrac{18}{5}\right)-\left(\left(\left(-4\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{7}\right)\sqrt{3}\right)\right)+\left(\dfrac{38}{3}\right)\sqrt{3}\\
&=&\left(-\dfrac{695}{21}\right)\sqrt{3}+\dfrac{47}{45}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{65}{3}\right)\sqrt{12}\right)-\dfrac{13}{3}+\dfrac{16}{9}+\dfrac{18}{5}\right)-\left(\left(\left(-2\right)\sqrt{12}\right)-\left(\left(-\dfrac{9}{7}\right)\sqrt{75}\right)\right)\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{12}\right)\\
&=&\left(\left(\left(\left(-\dfrac{130}{3}\right)\sqrt{3}\right)-\dfrac{13}{3}+\dfrac{16}{9}+\dfrac{18}{5}\right)-\left(\left(\left(-4\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{7}\right)\sqrt{3}\right)\right)\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{961}{21}\right)\sqrt{3}+\dfrac{47}{45}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{961}{21}\right)\sqrt{3}+\dfrac{47}{45}+\left(-\dfrac{38}{3}\right)\sqrt{3}\\
&=&\left(-\dfrac{409}{7}\right)\sqrt{3}+\dfrac{47}{45}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{65}{3}\right)\sqrt{12}\right)-\dfrac{13}{3}+\dfrac{16}{9}+\dfrac{18}{5}\right)-\left(\left(\left(-2\right)\sqrt{12}\right)-\left(\left(-\dfrac{9}{7}\right)\sqrt{75}\right)\right)\right)\times\left(\left(\dfrac{19}{3}\right)\sqrt{12}\right)\\
&=&\left(\left(\left(\left(-\dfrac{130}{3}\right)\sqrt{3}\right)-\dfrac{13}{3}+\dfrac{16}{9}+\dfrac{18}{5}\right)-\left(\left(\left(-4\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{7}\right)\sqrt{3}\right)\right)\right)\times\left(\left(\dfrac{38}{3}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{961}{21}\right)\sqrt{3}+\dfrac{47}{45}\right)\left(\left(\dfrac{38}{3}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{36518}{63}\right)\sqrt{9}+\left(\dfrac{1786}{135}\right)\sqrt{3}\\
&=&-\dfrac{36518}{21}+\left(\dfrac{1786}{135}\right)\sqrt{3}\\
\end{eqnarray*}