L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{74}{3}\right)\sqrt{75}\right)-\left(\left(-\dfrac{67}{9}\right)\sqrt{27}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{12}\right)\right)-\left(\left(5\right)\sqrt{9}\right)+5\) et \( Y=\left(\left(9\right)\sqrt{12}\right)-\left(\left(-\dfrac{27}{8}\right)\sqrt{75}+\left(\dfrac{35}{4}\right)\sqrt{12}+\left(\dfrac{8}{3}\right)\sqrt{75}+\left(-\dfrac{64}{9}\right)\sqrt{12}\right)\) . 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(\left(\dfrac{74}{3}\right)\sqrt{75}\right)-\left(\left(-\dfrac{67}{9}\right)\sqrt{27}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{12}\right)\right)-\left(\left(5\right)\sqrt{9}\right)+5\right)+\left(\left(\left(9\right)\sqrt{12}\right)-\left(\left(-\dfrac{27}{8}\right)\sqrt{75}+\left(\dfrac{35}{4}\right)\sqrt{12}+\left(\dfrac{8}{3}\right)\sqrt{75}+\left(-\dfrac{64}{9}\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{370}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{47}{3}\right)\sqrt{3}\right)\right)-15+5\right)+\left(\left(\left(18\right)\sqrt{3}\right)-\left(\left(-\dfrac{135}{8}\right)\sqrt{3}+\left(\dfrac{35}{2}\right)\sqrt{3}+\left(\dfrac{40}{3}\right)\sqrt{3}+\left(-\dfrac{128}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\left(\dfrac{370}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{47}{3}\right)\sqrt{3}\right)\right)-15+5+\left(\left(18\right)\sqrt{3}\right)-\left(\left(-\dfrac{135}{8}\right)\sqrt{3}+\left(\dfrac{35}{2}\right)\sqrt{3}+\left(\dfrac{40}{3}\right)\sqrt{3}+\left(-\dfrac{128}{9}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{10675}{72}\right)\sqrt{3}-10\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{74}{3}\right)\sqrt{75}\right)-\left(\left(-\dfrac{67}{9}\right)\sqrt{27}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{12}\right)\right)-\left(\left(5\right)\sqrt{9}\right)+5\right)-\left(\left(\left(9\right)\sqrt{12}\right)-\left(\left(-\dfrac{27}{8}\right)\sqrt{75}+\left(\dfrac{35}{4}\right)\sqrt{12}+\left(\dfrac{8}{3}\right)\sqrt{75}+\left(-\dfrac{64}{9}\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{370}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{47}{3}\right)\sqrt{3}\right)\right)-15+5\right)-\left(\left(\left(18\right)\sqrt{3}\right)-\left(\left(-\dfrac{135}{8}\right)\sqrt{3}+\left(\dfrac{35}{2}\right)\sqrt{3}+\left(\dfrac{40}{3}\right)\sqrt{3}+\left(-\dfrac{128}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(130\right)\sqrt{3}-10\right)-\left(\left(\dfrac{1315}{72}\right)\sqrt{3}\right)\\
&=&\left(130\right)\sqrt{3}-10+\left(-\dfrac{1315}{72}\right)\sqrt{3}\\
&=&\left(\dfrac{8045}{72}\right)\sqrt{3}-10\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{74}{3}\right)\sqrt{75}\right)-\left(\left(-\dfrac{67}{9}\right)\sqrt{27}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{12}\right)\right)-\left(\left(5\right)\sqrt{9}\right)+5\right)\times\left(\left(\left(9\right)\sqrt{12}\right)-\left(\left(-\dfrac{27}{8}\right)\sqrt{75}+\left(\dfrac{35}{4}\right)\sqrt{12}+\left(\dfrac{8}{3}\right)\sqrt{75}+\left(-\dfrac{64}{9}\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{370}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{47}{3}\right)\sqrt{3}\right)\right)-15+5\right)\times\left(\left(\left(18\right)\sqrt{3}\right)-\left(\left(-\dfrac{135}{8}\right)\sqrt{3}+\left(\dfrac{35}{2}\right)\sqrt{3}+\left(\dfrac{40}{3}\right)\sqrt{3}+\left(-\dfrac{128}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(130\right)\sqrt{3}-10\right)\left(\left(\dfrac{1315}{72}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{85475}{36}\right)\sqrt{9}+\left(-\dfrac{6575}{36}\right)\sqrt{3}\\
&=&\dfrac{85475}{12}+\left(-\dfrac{6575}{36}\right)\sqrt{3}\\
\end{eqnarray*}