L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{35}{9}\right)\sqrt{9}\right)-\dfrac{13}{7}-\left(\left(-\dfrac{35}{9}\right)\sqrt{9}\right)-\left(\left(-9\right)\sqrt{12}\right)+\dfrac{21}{4}-\left(\left(6\right)\sqrt{75}\right)+\dfrac{77}{3}+\left(-\dfrac{76}{9}\right)\sqrt{12}+\left(-\dfrac{53}{9}\right)\sqrt{27}+\left(-\dfrac{9}{7}\right)\sqrt{9}-\dfrac{26}{3}\) et \( Y=\left(\left(-8\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}+\dfrac{59}{3}+3-\dfrac{25}{2}+\left(-5\right)\sqrt{75}\right)-\left(\left(-3\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{43}{3}\right)\sqrt{12}\right)-\dfrac{7}{8}\right)\) . 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(-\dfrac{35}{9}\right)\sqrt{9}\right)-\dfrac{13}{7}-\left(\left(-\dfrac{35}{9}\right)\sqrt{9}\right)-\left(\left(-9\right)\sqrt{12}\right)+\dfrac{21}{4}-\left(\left(6\right)\sqrt{75}\right)+\dfrac{77}{3}+\left(-\dfrac{76}{9}\right)\sqrt{12}+\left(-\dfrac{53}{9}\right)\sqrt{27}+\left(-\dfrac{9}{7}\right)\sqrt{9}-\dfrac{26}{3}\right)+\left(\left(\left(-8\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}+\dfrac{59}{3}+3-\dfrac{25}{2}+\left(-5\right)\sqrt{75}\right)-\left(\left(-3\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{43}{3}\right)\sqrt{12}\right)-\dfrac{7}{8}\right)\right)\\
&=&\left(-\dfrac{35}{3}-\dfrac{13}{7}+\dfrac{35}{3}-\left(\left(-18\right)\sqrt{3}\right)+\dfrac{21}{4}-\left(\left(30\right)\sqrt{3}\right)+\dfrac{77}{3}+\left(-\dfrac{152}{9}\right)\sqrt{3}+\left(-\dfrac{53}{3}\right)\sqrt{3}-\dfrac{27}{7}-\dfrac{26}{3}\right)+\left(\left(\left(-24\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}+\dfrac{59}{3}+3-\dfrac{25}{2}+\left(-25\right)\sqrt{3}\right)+9-\left(\left(\left(\dfrac{86}{3}\right)\sqrt{3}\right)-\dfrac{7}{8}\right)\right)\\
&=&-\dfrac{35}{3}-\dfrac{13}{7}+\dfrac{35}{3}-\left(\left(-18\right)\sqrt{3}\right)+\dfrac{21}{4}-\left(\left(30\right)\sqrt{3}\right)+\dfrac{77}{3}+\left(-\dfrac{152}{9}\right)\sqrt{3}+\left(-\dfrac{53}{3}\right)\sqrt{3}-\dfrac{27}{7}-\dfrac{26}{3}+\left(\left(-24\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}+\dfrac{59}{3}+3-\dfrac{25}{2}+\left(-25\right)\sqrt{3}\right)+9-\left(\left(\left(\dfrac{86}{3}\right)\sqrt{3}\right)-\dfrac{7}{8}\right)\\
&=&\dfrac{2729}{168}+\left(-\dfrac{758}{9}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{35}{9}\right)\sqrt{9}\right)-\dfrac{13}{7}-\left(\left(-\dfrac{35}{9}\right)\sqrt{9}\right)-\left(\left(-9\right)\sqrt{12}\right)+\dfrac{21}{4}-\left(\left(6\right)\sqrt{75}\right)+\dfrac{77}{3}+\left(-\dfrac{76}{9}\right)\sqrt{12}+\left(-\dfrac{53}{9}\right)\sqrt{27}+\left(-\dfrac{9}{7}\right)\sqrt{9}-\dfrac{26}{3}\right)-\left(\left(\left(-8\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}+\dfrac{59}{3}+3-\dfrac{25}{2}+\left(-5\right)\sqrt{75}\right)-\left(\left(-3\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{43}{3}\right)\sqrt{12}\right)-\dfrac{7}{8}\right)\right)\\
&=&\left(-\dfrac{35}{3}-\dfrac{13}{7}+\dfrac{35}{3}-\left(\left(-18\right)\sqrt{3}\right)+\dfrac{21}{4}-\left(\left(30\right)\sqrt{3}\right)+\dfrac{77}{3}+\left(-\dfrac{152}{9}\right)\sqrt{3}+\left(-\dfrac{53}{3}\right)\sqrt{3}-\dfrac{27}{7}-\dfrac{26}{3}\right)-\left(\left(\left(-24\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}+\dfrac{59}{3}+3-\dfrac{25}{2}+\left(-25\right)\sqrt{3}\right)+9-\left(\left(\left(\dfrac{86}{3}\right)\sqrt{3}\right)-\dfrac{7}{8}\right)\right)\\
&=&\left(\dfrac{463}{28}+\left(-\dfrac{419}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{113}{3}\right)\sqrt{3}-\dfrac{7}{24}\right)\\
&=&\dfrac{463}{28}+\left(-\dfrac{419}{9}\right)\sqrt{3}+\left(\dfrac{113}{3}\right)\sqrt{3}+\dfrac{7}{24}\\
&=&\dfrac{2827}{168}+\left(-\dfrac{80}{9}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{35}{9}\right)\sqrt{9}\right)-\dfrac{13}{7}-\left(\left(-\dfrac{35}{9}\right)\sqrt{9}\right)-\left(\left(-9\right)\sqrt{12}\right)+\dfrac{21}{4}-\left(\left(6\right)\sqrt{75}\right)+\dfrac{77}{3}+\left(-\dfrac{76}{9}\right)\sqrt{12}+\left(-\dfrac{53}{9}\right)\sqrt{27}+\left(-\dfrac{9}{7}\right)\sqrt{9}-\dfrac{26}{3}\right)\times\left(\left(\left(-8\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}+\dfrac{59}{3}+3-\dfrac{25}{2}+\left(-5\right)\sqrt{75}\right)-\left(\left(-3\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{43}{3}\right)\sqrt{12}\right)-\dfrac{7}{8}\right)\right)\\
&=&\left(-\dfrac{35}{3}-\dfrac{13}{7}+\dfrac{35}{3}-\left(\left(-18\right)\sqrt{3}\right)+\dfrac{21}{4}-\left(\left(30\right)\sqrt{3}\right)+\dfrac{77}{3}+\left(-\dfrac{152}{9}\right)\sqrt{3}+\left(-\dfrac{53}{3}\right)\sqrt{3}-\dfrac{27}{7}-\dfrac{26}{3}\right)\times\left(\left(\left(-24\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}+\dfrac{59}{3}+3-\dfrac{25}{2}+\left(-25\right)\sqrt{3}\right)+9-\left(\left(\left(\dfrac{86}{3}\right)\sqrt{3}\right)-\dfrac{7}{8}\right)\right)\\
&=&\left(\dfrac{463}{28}+\left(-\dfrac{419}{9}\right)\sqrt{3}\right)\left(\left(-\dfrac{113}{3}\right)\sqrt{3}-\dfrac{7}{24}\right)\\
&=&\left(-\dfrac{921211}{1512}\right)\sqrt{3}-\dfrac{463}{96}+\left(\dfrac{47347}{27}\right)\sqrt{9}\\
&=&\left(-\dfrac{921211}{1512}\right)\sqrt{3}+\dfrac{1513715}{288}\\
\end{eqnarray*}