L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(2\right)\sqrt{12}\right)+4+\dfrac{27}{8}+\left(-2\right)\sqrt{27}-\dfrac{77}{5}+\left(-\dfrac{38}{9}\right)\sqrt{12}+\dfrac{63}{4}+\left(4\right)\sqrt{75}+\left(\left(1\right)\sqrt{27}\right)-\left(\left(-1\right)\sqrt{12}\right)+\dfrac{14}{9}-\left(\left(-\dfrac{17}{4}\right)\sqrt{9}\right)\) et \( Y=-\dfrac{23}{8}-\left(\left(-1\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{75}\right)+\left(0\right)\sqrt{9}+\left(-8\right)\sqrt{12}+\left(-\dfrac{59}{8}\right)\sqrt{9}+\left(-\dfrac{14}{9}\right)\sqrt{27}+\dfrac{21}{4}-\dfrac{3}{2}\) . 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(2\right)\sqrt{12}\right)+4+\dfrac{27}{8}+\left(-2\right)\sqrt{27}-\dfrac{77}{5}+\left(-\dfrac{38}{9}\right)\sqrt{12}+\dfrac{63}{4}+\left(4\right)\sqrt{75}+\left(\left(1\right)\sqrt{27}\right)-\left(\left(-1\right)\sqrt{12}\right)+\dfrac{14}{9}-\left(\left(-\dfrac{17}{4}\right)\sqrt{9}\right)\right)+\left(-\dfrac{23}{8}-\left(\left(-1\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{75}\right)+\left(0\right)\sqrt{9}+\left(-8\right)\sqrt{12}+\left(-\dfrac{59}{8}\right)\sqrt{9}+\left(-\dfrac{14}{9}\right)\sqrt{27}+\dfrac{21}{4}-\dfrac{3}{2}\right)\\
&=&\left(\left(\left(4\right)\sqrt{3}\right)+4+\dfrac{27}{8}+\left(-6\right)\sqrt{3}-\dfrac{77}{5}+\left(-\dfrac{76}{9}\right)\sqrt{3}+\dfrac{63}{4}+\left(20\right)\sqrt{3}+\left(\left(3\right)\sqrt{3}\right)-\left(\left(-2\right)\sqrt{3}\right)+\dfrac{14}{9}+\dfrac{51}{4}\right)+\left(-\dfrac{23}{8}-\left(\left(-2\right)\sqrt{3}\right)-\left(\left(-35\right)\sqrt{3}\right)+0+\left(-16\right)\sqrt{3}-\dfrac{177}{8}+\left(-\dfrac{14}{3}\right)\sqrt{3}+\dfrac{21}{4}-\dfrac{3}{2}\right)\\
&=&\left(\left(4\right)\sqrt{3}\right)+4+\dfrac{27}{8}+\left(-6\right)\sqrt{3}-\dfrac{77}{5}+\left(-\dfrac{76}{9}\right)\sqrt{3}+\dfrac{63}{4}+\left(20\right)\sqrt{3}+\left(\left(3\right)\sqrt{3}\right)-\left(\left(-2\right)\sqrt{3}\right)+\dfrac{14}{9}+\dfrac{51}{4}-\dfrac{23}{8}-\left(\left(-2\right)\sqrt{3}\right)-\left(\left(-35\right)\sqrt{3}\right)+0+\left(-16\right)\sqrt{3}-\dfrac{177}{8}+\left(-\dfrac{14}{3}\right)\sqrt{3}+\dfrac{21}{4}-\dfrac{3}{2}\\
&=&\left(\dfrac{278}{9}\right)\sqrt{3}+\dfrac{281}{360}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(2\right)\sqrt{12}\right)+4+\dfrac{27}{8}+\left(-2\right)\sqrt{27}-\dfrac{77}{5}+\left(-\dfrac{38}{9}\right)\sqrt{12}+\dfrac{63}{4}+\left(4\right)\sqrt{75}+\left(\left(1\right)\sqrt{27}\right)-\left(\left(-1\right)\sqrt{12}\right)+\dfrac{14}{9}-\left(\left(-\dfrac{17}{4}\right)\sqrt{9}\right)\right)-\left(-\dfrac{23}{8}-\left(\left(-1\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{75}\right)+\left(0\right)\sqrt{9}+\left(-8\right)\sqrt{12}+\left(-\dfrac{59}{8}\right)\sqrt{9}+\left(-\dfrac{14}{9}\right)\sqrt{27}+\dfrac{21}{4}-\dfrac{3}{2}\right)\\
&=&\left(\left(\left(4\right)\sqrt{3}\right)+4+\dfrac{27}{8}+\left(-6\right)\sqrt{3}-\dfrac{77}{5}+\left(-\dfrac{76}{9}\right)\sqrt{3}+\dfrac{63}{4}+\left(20\right)\sqrt{3}+\left(\left(3\right)\sqrt{3}\right)-\left(\left(-2\right)\sqrt{3}\right)+\dfrac{14}{9}+\dfrac{51}{4}\right)-\left(-\dfrac{23}{8}-\left(\left(-2\right)\sqrt{3}\right)-\left(\left(-35\right)\sqrt{3}\right)+0+\left(-16\right)\sqrt{3}-\dfrac{177}{8}+\left(-\dfrac{14}{3}\right)\sqrt{3}+\dfrac{21}{4}-\dfrac{3}{2}\right)\\
&=&\left(\left(\dfrac{131}{9}\right)\sqrt{3}+\dfrac{7931}{360}\right)-\left(-\dfrac{85}{4}+\left(\dfrac{49}{3}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{131}{9}\right)\sqrt{3}+\dfrac{7931}{360}+\dfrac{85}{4}+\left(-\dfrac{49}{3}\right)\sqrt{3}\\
&=&\left(-\dfrac{16}{9}\right)\sqrt{3}+\dfrac{15581}{360}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(2\right)\sqrt{12}\right)+4+\dfrac{27}{8}+\left(-2\right)\sqrt{27}-\dfrac{77}{5}+\left(-\dfrac{38}{9}\right)\sqrt{12}+\dfrac{63}{4}+\left(4\right)\sqrt{75}+\left(\left(1\right)\sqrt{27}\right)-\left(\left(-1\right)\sqrt{12}\right)+\dfrac{14}{9}-\left(\left(-\dfrac{17}{4}\right)\sqrt{9}\right)\right)\times\left(-\dfrac{23}{8}-\left(\left(-1\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{75}\right)+\left(0\right)\sqrt{9}+\left(-8\right)\sqrt{12}+\left(-\dfrac{59}{8}\right)\sqrt{9}+\left(-\dfrac{14}{9}\right)\sqrt{27}+\dfrac{21}{4}-\dfrac{3}{2}\right)\\
&=&\left(\left(\left(4\right)\sqrt{3}\right)+4+\dfrac{27}{8}+\left(-6\right)\sqrt{3}-\dfrac{77}{5}+\left(-\dfrac{76}{9}\right)\sqrt{3}+\dfrac{63}{4}+\left(20\right)\sqrt{3}+\left(\left(3\right)\sqrt{3}\right)-\left(\left(-2\right)\sqrt{3}\right)+\dfrac{14}{9}+\dfrac{51}{4}\right)\times\left(-\dfrac{23}{8}-\left(\left(-2\right)\sqrt{3}\right)-\left(\left(-35\right)\sqrt{3}\right)+0+\left(-16\right)\sqrt{3}-\dfrac{177}{8}+\left(-\dfrac{14}{3}\right)\sqrt{3}+\dfrac{21}{4}-\dfrac{3}{2}\right)\\
&=&\left(\left(\dfrac{131}{9}\right)\sqrt{3}+\dfrac{7931}{360}\right)\left(-\dfrac{85}{4}+\left(\dfrac{49}{3}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{54569}{1080}\right)\sqrt{3}+\left(\dfrac{6419}{27}\right)\sqrt{9}-\dfrac{134827}{288}\\
&=&\left(\dfrac{54569}{1080}\right)\sqrt{3}+\dfrac{23527}{96}\\
\end{eqnarray*}