L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{46}{7}\right)\sqrt{8}-\dfrac{45}{4}-9+\left(-\dfrac{35}{6}\right)\sqrt{8}+\left(\dfrac{5}{9}\right)\sqrt{4}+\left(-4\right)\sqrt{8}+\left(7\right)\sqrt{18}-4+\dfrac{1}{3}+\left(0\right)\sqrt{4}+\left(5\right)\sqrt{50}\) et \( Y=\left(\left(-\dfrac{65}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{41}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{53}{7}\right)\sqrt{8}\right)-\left(-\dfrac{61}{8}-\left(\left(6\right)\sqrt{18}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{46}{7}\right)\sqrt{8}-\dfrac{45}{4}-9+\left(-\dfrac{35}{6}\right)\sqrt{8}+\left(\dfrac{5}{9}\right)\sqrt{4}+\left(-4\right)\sqrt{8}+\left(7\right)\sqrt{18}-4+\dfrac{1}{3}+\left(0\right)\sqrt{4}+\left(5\right)\sqrt{50}\right)+\left(\left(\left(-\dfrac{65}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{41}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{53}{7}\right)\sqrt{8}\right)-\left(-\dfrac{61}{8}-\left(\left(6\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\left(\dfrac{92}{7}\right)\sqrt{2}-\dfrac{45}{4}-9+\left(-\dfrac{35}{3}\right)\sqrt{2}+\dfrac{10}{9}+\left(-8\right)\sqrt{2}+\left(21\right)\sqrt{2}-4+\dfrac{1}{3}+0+\left(25\right)\sqrt{2}\right)+\left(-\dfrac{130}{9}-\dfrac{82}{9}-\left(\left(\dfrac{106}{7}\right)\sqrt{2}\right)-\left(-\dfrac{61}{8}-\left(\left(18\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\dfrac{92}{7}\right)\sqrt{2}-\dfrac{45}{4}-9+\left(-\dfrac{35}{3}\right)\sqrt{2}+\dfrac{10}{9}+\left(-8\right)\sqrt{2}+\left(21\right)\sqrt{2}-4+\dfrac{1}{3}+0+\left(25\right)\sqrt{2}-\dfrac{130}{9}-\dfrac{82}{9}-\left(\left(\dfrac{106}{7}\right)\sqrt{2}\right)-\left(-\dfrac{61}{8}-\left(\left(18\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{127}{3}\right)\sqrt{2}-\dfrac{2789}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{46}{7}\right)\sqrt{8}-\dfrac{45}{4}-9+\left(-\dfrac{35}{6}\right)\sqrt{8}+\left(\dfrac{5}{9}\right)\sqrt{4}+\left(-4\right)\sqrt{8}+\left(7\right)\sqrt{18}-4+\dfrac{1}{3}+\left(0\right)\sqrt{4}+\left(5\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{65}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{41}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{53}{7}\right)\sqrt{8}\right)-\left(-\dfrac{61}{8}-\left(\left(6\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\left(\dfrac{92}{7}\right)\sqrt{2}-\dfrac{45}{4}-9+\left(-\dfrac{35}{3}\right)\sqrt{2}+\dfrac{10}{9}+\left(-8\right)\sqrt{2}+\left(21\right)\sqrt{2}-4+\dfrac{1}{3}+0+\left(25\right)\sqrt{2}\right)-\left(-\dfrac{130}{9}-\dfrac{82}{9}-\left(\left(\dfrac{106}{7}\right)\sqrt{2}\right)-\left(-\dfrac{61}{8}-\left(\left(18\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(\dfrac{829}{21}\right)\sqrt{2}-\dfrac{821}{36}\right)-\left(-\dfrac{1147}{72}+\left(\dfrac{20}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{829}{21}\right)\sqrt{2}-\dfrac{821}{36}+\dfrac{1147}{72}+\left(-\dfrac{20}{7}\right)\sqrt{2}\\
&=&\left(\dfrac{769}{21}\right)\sqrt{2}-\dfrac{55}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{46}{7}\right)\sqrt{8}-\dfrac{45}{4}-9+\left(-\dfrac{35}{6}\right)\sqrt{8}+\left(\dfrac{5}{9}\right)\sqrt{4}+\left(-4\right)\sqrt{8}+\left(7\right)\sqrt{18}-4+\dfrac{1}{3}+\left(0\right)\sqrt{4}+\left(5\right)\sqrt{50}\right)\times\left(\left(\left(-\dfrac{65}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{41}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{53}{7}\right)\sqrt{8}\right)-\left(-\dfrac{61}{8}-\left(\left(6\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\left(\dfrac{92}{7}\right)\sqrt{2}-\dfrac{45}{4}-9+\left(-\dfrac{35}{3}\right)\sqrt{2}+\dfrac{10}{9}+\left(-8\right)\sqrt{2}+\left(21\right)\sqrt{2}-4+\dfrac{1}{3}+0+\left(25\right)\sqrt{2}\right)\times\left(-\dfrac{130}{9}-\dfrac{82}{9}-\left(\left(\dfrac{106}{7}\right)\sqrt{2}\right)-\left(-\dfrac{61}{8}-\left(\left(18\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(\dfrac{829}{21}\right)\sqrt{2}-\dfrac{821}{36}\right)\left(-\dfrac{1147}{72}+\left(\dfrac{20}{7}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{1049383}{1512}\right)\sqrt{2}+\left(\dfrac{16580}{147}\right)\sqrt{4}+\dfrac{941687}{2592}\\
&=&\left(-\dfrac{1049383}{1512}\right)\sqrt{2}+\dfrac{74792903}{127008}\\
\end{eqnarray*}