L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(1\right)\sqrt{27}\right)+\dfrac{71}{5}+\dfrac{41}{6}-\left(\left(-\dfrac{44}{3}\right)\sqrt{75}\right)\) et \( Y=\dfrac{16}{3}-\left(\left(\left(5\right)\sqrt{12}\right)-\left(\left(7\right)\sqrt{75}\right)-\left(\left(-9\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{9}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(1\right)\sqrt{27}\right)+\dfrac{71}{5}+\dfrac{41}{6}-\left(\left(-\dfrac{44}{3}\right)\sqrt{75}\right)\right)+\left(\dfrac{16}{3}-\left(\left(\left(5\right)\sqrt{12}\right)-\left(\left(7\right)\sqrt{75}\right)-\left(\left(-9\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(\left(3\right)\sqrt{3}\right)+\dfrac{71}{5}+\dfrac{41}{6}-\left(\left(-\dfrac{220}{3}\right)\sqrt{3}\right)\right)+\left(\dfrac{16}{3}-\left(\left(\left(10\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)+27-21\right)\right)\\
&=&\left(\left(3\right)\sqrt{3}\right)+\dfrac{71}{5}+\dfrac{41}{6}-\left(\left(-\dfrac{220}{3}\right)\sqrt{3}\right)+\dfrac{16}{3}-\left(\left(\left(10\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)+27-21\right)\\
&=&\left(\dfrac{304}{3}\right)\sqrt{3}+\dfrac{611}{30}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(1\right)\sqrt{27}\right)+\dfrac{71}{5}+\dfrac{41}{6}-\left(\left(-\dfrac{44}{3}\right)\sqrt{75}\right)\right)-\left(\dfrac{16}{3}-\left(\left(\left(5\right)\sqrt{12}\right)-\left(\left(7\right)\sqrt{75}\right)-\left(\left(-9\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(\left(3\right)\sqrt{3}\right)+\dfrac{71}{5}+\dfrac{41}{6}-\left(\left(-\dfrac{220}{3}\right)\sqrt{3}\right)\right)-\left(\dfrac{16}{3}-\left(\left(\left(10\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)+27-21\right)\right)\\
&=&\left(\left(\dfrac{229}{3}\right)\sqrt{3}+\dfrac{631}{30}\right)-\left(-\dfrac{2}{3}+\left(25\right)\sqrt{3}\right)\\
&=&\left(\dfrac{229}{3}\right)\sqrt{3}+\dfrac{631}{30}+\dfrac{2}{3}+\left(-25\right)\sqrt{3}\\
&=&\left(\dfrac{154}{3}\right)\sqrt{3}+\dfrac{217}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(1\right)\sqrt{27}\right)+\dfrac{71}{5}+\dfrac{41}{6}-\left(\left(-\dfrac{44}{3}\right)\sqrt{75}\right)\right)\times\left(\dfrac{16}{3}-\left(\left(\left(5\right)\sqrt{12}\right)-\left(\left(7\right)\sqrt{75}\right)-\left(\left(-9\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(\left(3\right)\sqrt{3}\right)+\dfrac{71}{5}+\dfrac{41}{6}-\left(\left(-\dfrac{220}{3}\right)\sqrt{3}\right)\right)\times\left(\dfrac{16}{3}-\left(\left(\left(10\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)+27-21\right)\right)\\
&=&\left(\left(\dfrac{229}{3}\right)\sqrt{3}+\dfrac{631}{30}\right)\left(-\dfrac{2}{3}+\left(25\right)\sqrt{3}\right)\\
&=&\left(\dfrac{8549}{18}\right)\sqrt{3}+\left(\dfrac{5725}{3}\right)\sqrt{9}-\dfrac{631}{45}\\
&=&\left(\dfrac{8549}{18}\right)\sqrt{3}+\dfrac{256994}{45}\\
\end{eqnarray*}