L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{35}{6}\right)\sqrt{9}\right)-2-\left(\left(3\right)\sqrt{9}\right)\right)-\left(\left(-3\right)\sqrt{27}-\dfrac{55}{2}+\left(3\right)\sqrt{9}+\left(-\dfrac{80}{9}\right)\sqrt{27}\right)\) et \( Y=\left(-7-\left(\left(5\right)\sqrt{9}\right)-\left(\left(0\right)\sqrt{12}\right)-\dfrac{39}{2}-\left(\left(\dfrac{17}{4}\right)\sqrt{12}\right)\right)-\left(\left(0\right)\sqrt{12}\right)\) . 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{35}{6}\right)\sqrt{9}\right)-2-\left(\left(3\right)\sqrt{9}\right)\right)-\left(\left(-3\right)\sqrt{27}-\dfrac{55}{2}+\left(3\right)\sqrt{9}+\left(-\dfrac{80}{9}\right)\sqrt{27}\right)\right)+\left(\left(-7-\left(\left(5\right)\sqrt{9}\right)-\left(\left(0\right)\sqrt{12}\right)-\dfrac{39}{2}-\left(\left(\dfrac{17}{4}\right)\sqrt{12}\right)\right)-\left(\left(0\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\dfrac{35}{2}-2-9\right)-\left(\left(-9\right)\sqrt{3}-\dfrac{55}{2}+9+\left(-\dfrac{80}{3}\right)\sqrt{3}\right)\right)+\left(\left(-7-15-\left(\left(0\right)\sqrt{3}\right)-\dfrac{39}{2}-\left(\left(\dfrac{17}{2}\right)\sqrt{3}\right)\right)-\left(\left(0\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{35}{2}-2-9\right)-\left(\left(-9\right)\sqrt{3}-\dfrac{55}{2}+9+\left(-\dfrac{80}{3}\right)\sqrt{3}\right)+\left(-7-15-\left(\left(0\right)\sqrt{3}\right)-\dfrac{39}{2}-\left(\left(\dfrac{17}{2}\right)\sqrt{3}\right)\right)-\left(\left(0\right)\sqrt{3}\right)\\
&=&-\dfrac{33}{2}+\left(\dfrac{163}{6}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{35}{6}\right)\sqrt{9}\right)-2-\left(\left(3\right)\sqrt{9}\right)\right)-\left(\left(-3\right)\sqrt{27}-\dfrac{55}{2}+\left(3\right)\sqrt{9}+\left(-\dfrac{80}{9}\right)\sqrt{27}\right)\right)-\left(\left(-7-\left(\left(5\right)\sqrt{9}\right)-\left(\left(0\right)\sqrt{12}\right)-\dfrac{39}{2}-\left(\left(\dfrac{17}{4}\right)\sqrt{12}\right)\right)-\left(\left(0\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\dfrac{35}{2}-2-9\right)-\left(\left(-9\right)\sqrt{3}-\dfrac{55}{2}+9+\left(-\dfrac{80}{3}\right)\sqrt{3}\right)\right)-\left(\left(-7-15-\left(\left(0\right)\sqrt{3}\right)-\dfrac{39}{2}-\left(\left(\dfrac{17}{2}\right)\sqrt{3}\right)\right)-\left(\left(0\right)\sqrt{3}\right)\right)\\
&=&\left(25+\left(\dfrac{107}{3}\right)\sqrt{3}\right)-\left(-\dfrac{83}{2}+\left(-\dfrac{17}{2}\right)\sqrt{3}\right)\\
&=&25+\left(\dfrac{107}{3}\right)\sqrt{3}+\dfrac{83}{2}+\left(\dfrac{17}{2}\right)\sqrt{3}\\
&=&\dfrac{133}{2}+\left(\dfrac{265}{6}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{35}{6}\right)\sqrt{9}\right)-2-\left(\left(3\right)\sqrt{9}\right)\right)-\left(\left(-3\right)\sqrt{27}-\dfrac{55}{2}+\left(3\right)\sqrt{9}+\left(-\dfrac{80}{9}\right)\sqrt{27}\right)\right)\times\left(\left(-7-\left(\left(5\right)\sqrt{9}\right)-\left(\left(0\right)\sqrt{12}\right)-\dfrac{39}{2}-\left(\left(\dfrac{17}{4}\right)\sqrt{12}\right)\right)-\left(\left(0\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\dfrac{35}{2}-2-9\right)-\left(\left(-9\right)\sqrt{3}-\dfrac{55}{2}+9+\left(-\dfrac{80}{3}\right)\sqrt{3}\right)\right)\times\left(\left(-7-15-\left(\left(0\right)\sqrt{3}\right)-\dfrac{39}{2}-\left(\left(\dfrac{17}{2}\right)\sqrt{3}\right)\right)-\left(\left(0\right)\sqrt{3}\right)\right)\\
&=&\left(25+\left(\dfrac{107}{3}\right)\sqrt{3}\right)\left(-\dfrac{83}{2}+\left(-\dfrac{17}{2}\right)\sqrt{3}\right)\\
&=&-\dfrac{2075}{2}+\left(-\dfrac{5078}{3}\right)\sqrt{3}+\left(-\dfrac{1819}{6}\right)\sqrt{9}\\
&=&-1947+\left(-\dfrac{5078}{3}\right)\sqrt{3}\\
\end{eqnarray*}