L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{19}{3}\right)\sqrt{50}\) et \( Y=\left(-\dfrac{79}{9}\right)\sqrt{4}+\left(\dfrac{44}{9}\right)\sqrt{50}+\left(\dfrac{79}{6}\right)\sqrt{50}+\left(\dfrac{25}{3}\right)\sqrt{18}+\left(0\right)\sqrt{18}\) . 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{19}{3}\right)\sqrt{50}\right)+\left(\left(-\dfrac{79}{9}\right)\sqrt{4}+\left(\dfrac{44}{9}\right)\sqrt{50}+\left(\dfrac{79}{6}\right)\sqrt{50}+\left(\dfrac{25}{3}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)\\
&=&\left(\left(-\dfrac{95}{3}\right)\sqrt{2}\right)+\left(-\dfrac{158}{9}+\left(\dfrac{220}{9}\right)\sqrt{2}+\left(\dfrac{395}{6}\right)\sqrt{2}+\left(25\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{95}{3}\right)\sqrt{2}-\dfrac{158}{9}+\left(\dfrac{220}{9}\right)\sqrt{2}+\left(\dfrac{395}{6}\right)\sqrt{2}+\left(25\right)\sqrt{2}+\left(0\right)\sqrt{2}\\
&=&\left(\dfrac{1505}{18}\right)\sqrt{2}-\dfrac{158}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{19}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{79}{9}\right)\sqrt{4}+\left(\dfrac{44}{9}\right)\sqrt{50}+\left(\dfrac{79}{6}\right)\sqrt{50}+\left(\dfrac{25}{3}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)\\
&=&\left(\left(-\dfrac{95}{3}\right)\sqrt{2}\right)-\left(-\dfrac{158}{9}+\left(\dfrac{220}{9}\right)\sqrt{2}+\left(\dfrac{395}{6}\right)\sqrt{2}+\left(25\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{95}{3}\right)\sqrt{2}\right)-\left(-\dfrac{158}{9}+\left(\dfrac{2075}{18}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{95}{3}\right)\sqrt{2}+\dfrac{158}{9}+\left(-\dfrac{2075}{18}\right)\sqrt{2}\\
&=&\left(-\dfrac{2645}{18}\right)\sqrt{2}+\dfrac{158}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{19}{3}\right)\sqrt{50}\right)\times\left(\left(-\dfrac{79}{9}\right)\sqrt{4}+\left(\dfrac{44}{9}\right)\sqrt{50}+\left(\dfrac{79}{6}\right)\sqrt{50}+\left(\dfrac{25}{3}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)\\
&=&\left(\left(-\dfrac{95}{3}\right)\sqrt{2}\right)\times\left(-\dfrac{158}{9}+\left(\dfrac{220}{9}\right)\sqrt{2}+\left(\dfrac{395}{6}\right)\sqrt{2}+\left(25\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{95}{3}\right)\sqrt{2}\right)\left(-\dfrac{158}{9}+\left(\dfrac{2075}{18}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{15010}{27}\right)\sqrt{2}+\left(-\dfrac{197125}{54}\right)\sqrt{4}\\
&=&\left(\dfrac{15010}{27}\right)\sqrt{2}-\dfrac{197125}{27}\\
\end{eqnarray*}