L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=4+\left(4\right)\sqrt{50}+\left(-\dfrac{73}{2}\right)\sqrt{8}+\left(-\dfrac{31}{4}\right)\sqrt{8}+\left(8\right)\sqrt{8}+\left(-\dfrac{11}{6}\right)\sqrt{18}+\left(2\right)\sqrt{50}-\dfrac{59}{8}-8+\dfrac{62}{5}\) et \( Y=\left(-2\right)\sqrt{8}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(4+\left(4\right)\sqrt{50}+\left(-\dfrac{73}{2}\right)\sqrt{8}+\left(-\dfrac{31}{4}\right)\sqrt{8}+\left(8\right)\sqrt{8}+\left(-\dfrac{11}{6}\right)\sqrt{18}+\left(2\right)\sqrt{50}-\dfrac{59}{8}-8+\dfrac{62}{5}\right)+\left(\left(-2\right)\sqrt{8}\right)\\
&=&\left(4+\left(20\right)\sqrt{2}+\left(-73\right)\sqrt{2}+\left(-\dfrac{31}{2}\right)\sqrt{2}+\left(16\right)\sqrt{2}+\left(-\dfrac{11}{2}\right)\sqrt{2}+\left(10\right)\sqrt{2}-\dfrac{59}{8}-8+\dfrac{62}{5}\right)+\left(\left(-4\right)\sqrt{2}\right)\\
&=&4+\left(20\right)\sqrt{2}+\left(-73\right)\sqrt{2}+\left(-\dfrac{31}{2}\right)\sqrt{2}+\left(16\right)\sqrt{2}+\left(-\dfrac{11}{2}\right)\sqrt{2}+\left(10\right)\sqrt{2}-\dfrac{59}{8}-8+\dfrac{62}{5}+\left(-4\right)\sqrt{2}\\
&=&\dfrac{41}{40}+\left(-52\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(4+\left(4\right)\sqrt{50}+\left(-\dfrac{73}{2}\right)\sqrt{8}+\left(-\dfrac{31}{4}\right)\sqrt{8}+\left(8\right)\sqrt{8}+\left(-\dfrac{11}{6}\right)\sqrt{18}+\left(2\right)\sqrt{50}-\dfrac{59}{8}-8+\dfrac{62}{5}\right)-\left(\left(-2\right)\sqrt{8}\right)\\
&=&\left(4+\left(20\right)\sqrt{2}+\left(-73\right)\sqrt{2}+\left(-\dfrac{31}{2}\right)\sqrt{2}+\left(16\right)\sqrt{2}+\left(-\dfrac{11}{2}\right)\sqrt{2}+\left(10\right)\sqrt{2}-\dfrac{59}{8}-8+\dfrac{62}{5}\right)-\left(\left(-4\right)\sqrt{2}\right)\\
&=&\left(\dfrac{41}{40}+\left(-48\right)\sqrt{2}\right)-\left(\left(-4\right)\sqrt{2}\right)\\
&=&\dfrac{41}{40}+\left(-48\right)\sqrt{2}+\left(4\right)\sqrt{2}\\
&=&\dfrac{41}{40}+\left(-44\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(4+\left(4\right)\sqrt{50}+\left(-\dfrac{73}{2}\right)\sqrt{8}+\left(-\dfrac{31}{4}\right)\sqrt{8}+\left(8\right)\sqrt{8}+\left(-\dfrac{11}{6}\right)\sqrt{18}+\left(2\right)\sqrt{50}-\dfrac{59}{8}-8+\dfrac{62}{5}\right)\times\left(\left(-2\right)\sqrt{8}\right)\\
&=&\left(4+\left(20\right)\sqrt{2}+\left(-73\right)\sqrt{2}+\left(-\dfrac{31}{2}\right)\sqrt{2}+\left(16\right)\sqrt{2}+\left(-\dfrac{11}{2}\right)\sqrt{2}+\left(10\right)\sqrt{2}-\dfrac{59}{8}-8+\dfrac{62}{5}\right)\times\left(\left(-4\right)\sqrt{2}\right)\\
&=&\left(\dfrac{41}{40}+\left(-48\right)\sqrt{2}\right)\left(\left(-4\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{41}{10}\right)\sqrt{2}+\left(192\right)\sqrt{4}\\
&=&\left(-\dfrac{41}{10}\right)\sqrt{2}+384\\
\end{eqnarray*}