L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-3\right)\sqrt{8}\right)-\left(\left(-\dfrac{79}{3}\right)\sqrt{50}+\dfrac{22}{3}-4\right)\) et \( Y=\left(\left(\dfrac{35}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{36}{5}\right)\sqrt{8}+\left(-\dfrac{19}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{8}\right)\) . 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(-3\right)\sqrt{8}\right)-\left(\left(-\dfrac{79}{3}\right)\sqrt{50}+\dfrac{22}{3}-4\right)\right)+\left(\left(\left(\dfrac{35}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{36}{5}\right)\sqrt{8}+\left(-\dfrac{19}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{8}\right)\right)\\
&=&\left(\left(\left(-6\right)\sqrt{2}\right)-\left(\left(-\dfrac{395}{3}\right)\sqrt{2}+\dfrac{22}{3}-4\right)\right)+\left(\dfrac{70}{9}-\left(\left(\dfrac{72}{5}\right)\sqrt{2}-19\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-6\right)\sqrt{2}\right)-\left(\left(-\dfrac{395}{3}\right)\sqrt{2}+\dfrac{22}{3}-4\right)+\dfrac{70}{9}-\left(\left(\dfrac{72}{5}\right)\sqrt{2}-19\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{473}{5}\right)\sqrt{2}+\dfrac{211}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-3\right)\sqrt{8}\right)-\left(\left(-\dfrac{79}{3}\right)\sqrt{50}+\dfrac{22}{3}-4\right)\right)-\left(\left(\left(\dfrac{35}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{36}{5}\right)\sqrt{8}+\left(-\dfrac{19}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{8}\right)\right)\\
&=&\left(\left(\left(-6\right)\sqrt{2}\right)-\left(\left(-\dfrac{395}{3}\right)\sqrt{2}+\dfrac{22}{3}-4\right)\right)-\left(\dfrac{70}{9}-\left(\left(\dfrac{72}{5}\right)\sqrt{2}-19\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{377}{3}\right)\sqrt{2}-\dfrac{10}{3}\right)-\left(\dfrac{241}{9}+\left(-\dfrac{466}{15}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{377}{3}\right)\sqrt{2}-\dfrac{10}{3}+-\dfrac{241}{9}+\left(\dfrac{466}{15}\right)\sqrt{2}\\
&=&\left(\dfrac{2351}{15}\right)\sqrt{2}-\dfrac{271}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-3\right)\sqrt{8}\right)-\left(\left(-\dfrac{79}{3}\right)\sqrt{50}+\dfrac{22}{3}-4\right)\right)\times\left(\left(\left(\dfrac{35}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{36}{5}\right)\sqrt{8}+\left(-\dfrac{19}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{8}\right)\right)\\
&=&\left(\left(\left(-6\right)\sqrt{2}\right)-\left(\left(-\dfrac{395}{3}\right)\sqrt{2}+\dfrac{22}{3}-4\right)\right)\times\left(\dfrac{70}{9}-\left(\left(\dfrac{72}{5}\right)\sqrt{2}-19\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{377}{3}\right)\sqrt{2}-\dfrac{10}{3}\right)\left(\dfrac{241}{9}+\left(-\dfrac{466}{15}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{93653}{27}\right)\sqrt{2}+\left(-\dfrac{175682}{45}\right)\sqrt{4}-\dfrac{2410}{27}\\
&=&\left(\dfrac{93653}{27}\right)\sqrt{2}-\dfrac{1066142}{135}\\
\end{eqnarray*}