L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(0\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{9}\right)-\left(\left(4\right)\sqrt{75}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{9}\right)+\dfrac{53}{5}+\left(-\dfrac{1}{9}\right)\sqrt{12}+\left(-6\right)\sqrt{27}+\left(3\right)\sqrt{12}\) et \( Y=-\dfrac{35}{3}+\left(\dfrac{11}{5}\right)\sqrt{12}+\left(\dfrac{54}{5}\right)\sqrt{27}\) . 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(0\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{9}\right)-\left(\left(4\right)\sqrt{75}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{9}\right)+\dfrac{53}{5}+\left(-\dfrac{1}{9}\right)\sqrt{12}+\left(-6\right)\sqrt{27}+\left(3\right)\sqrt{12}\right)+\left(-\dfrac{35}{3}+\left(\dfrac{11}{5}\right)\sqrt{12}+\left(\dfrac{54}{5}\right)\sqrt{27}\right)\\
&=&\left(\left(\left(0\right)\sqrt{3}\right)+\dfrac{3}{2}-\left(\left(20\right)\sqrt{3}\right)+\dfrac{51}{2}+\dfrac{53}{5}+\left(-\dfrac{2}{9}\right)\sqrt{3}+\left(-18\right)\sqrt{3}+\left(6\right)\sqrt{3}\right)+\left(-\dfrac{35}{3}+\left(\dfrac{22}{5}\right)\sqrt{3}+\left(\dfrac{162}{5}\right)\sqrt{3}\right)\\
&=&\left(\left(0\right)\sqrt{3}\right)+\dfrac{3}{2}-\left(\left(20\right)\sqrt{3}\right)+\dfrac{51}{2}+\dfrac{53}{5}+\left(-\dfrac{2}{9}\right)\sqrt{3}+\left(-18\right)\sqrt{3}+\left(6\right)\sqrt{3}-\dfrac{35}{3}+\left(\dfrac{22}{5}\right)\sqrt{3}+\left(\dfrac{162}{5}\right)\sqrt{3}\\
&=&\left(\dfrac{206}{45}\right)\sqrt{3}+\dfrac{389}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(0\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{9}\right)-\left(\left(4\right)\sqrt{75}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{9}\right)+\dfrac{53}{5}+\left(-\dfrac{1}{9}\right)\sqrt{12}+\left(-6\right)\sqrt{27}+\left(3\right)\sqrt{12}\right)-\left(-\dfrac{35}{3}+\left(\dfrac{11}{5}\right)\sqrt{12}+\left(\dfrac{54}{5}\right)\sqrt{27}\right)\\
&=&\left(\left(\left(0\right)\sqrt{3}\right)+\dfrac{3}{2}-\left(\left(20\right)\sqrt{3}\right)+\dfrac{51}{2}+\dfrac{53}{5}+\left(-\dfrac{2}{9}\right)\sqrt{3}+\left(-18\right)\sqrt{3}+\left(6\right)\sqrt{3}\right)-\left(-\dfrac{35}{3}+\left(\dfrac{22}{5}\right)\sqrt{3}+\left(\dfrac{162}{5}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{290}{9}\right)\sqrt{3}+\dfrac{188}{5}\right)-\left(-\dfrac{35}{3}+\left(\dfrac{184}{5}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{290}{9}\right)\sqrt{3}+\dfrac{188}{5}+\dfrac{35}{3}+\left(-\dfrac{184}{5}\right)\sqrt{3}\\
&=&\left(-\dfrac{3106}{45}\right)\sqrt{3}+\dfrac{739}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(0\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{9}\right)-\left(\left(4\right)\sqrt{75}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{9}\right)+\dfrac{53}{5}+\left(-\dfrac{1}{9}\right)\sqrt{12}+\left(-6\right)\sqrt{27}+\left(3\right)\sqrt{12}\right)\times\left(-\dfrac{35}{3}+\left(\dfrac{11}{5}\right)\sqrt{12}+\left(\dfrac{54}{5}\right)\sqrt{27}\right)\\
&=&\left(\left(\left(0\right)\sqrt{3}\right)+\dfrac{3}{2}-\left(\left(20\right)\sqrt{3}\right)+\dfrac{51}{2}+\dfrac{53}{5}+\left(-\dfrac{2}{9}\right)\sqrt{3}+\left(-18\right)\sqrt{3}+\left(6\right)\sqrt{3}\right)\times\left(-\dfrac{35}{3}+\left(\dfrac{22}{5}\right)\sqrt{3}+\left(\dfrac{162}{5}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{290}{9}\right)\sqrt{3}+\dfrac{188}{5}\right)\left(-\dfrac{35}{3}+\left(\dfrac{184}{5}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{1187734}{675}\right)\sqrt{3}+\left(-\dfrac{10672}{9}\right)\sqrt{9}-\dfrac{1316}{3}\\
&=&\left(\dfrac{1187734}{675}\right)\sqrt{3}-3996\\
\end{eqnarray*}