L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{14}{5}-\left(\dfrac{20}{3}+\left(-\dfrac{35}{4}\right)\sqrt{50}+\left(-9\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{29}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{18}\right)-\left(\left(\dfrac{2}{7}\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{53}{3}\right)\sqrt{4}\right)\) et \( Y=\left(\dfrac{50}{3}\right)\sqrt{4}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{14}{5}-\left(\dfrac{20}{3}+\left(-\dfrac{35}{4}\right)\sqrt{50}+\left(-9\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{29}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{18}\right)-\left(\left(\dfrac{2}{7}\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{53}{3}\right)\sqrt{4}\right)\right)+\left(\left(\dfrac{50}{3}\right)\sqrt{4}\right)\\
&=&\left(\dfrac{14}{5}-\left(\dfrac{20}{3}+\left(-\dfrac{175}{4}\right)\sqrt{2}+\left(-27\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{87}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{135}{8}\right)\sqrt{2}\right)-\dfrac{4}{7}\right)-\dfrac{106}{3}\right)+\left(\dfrac{100}{3}\right)\\
&=&\dfrac{14}{5}-\left(\dfrac{20}{3}+\left(-\dfrac{175}{4}\right)\sqrt{2}+\left(-27\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{87}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{135}{8}\right)\sqrt{2}\right)-\dfrac{4}{7}\right)-\dfrac{106}{3}+\dfrac{100}{3}\\
&=&-\dfrac{556}{105}+\left(\dfrac{527}{8}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{14}{5}-\left(\dfrac{20}{3}+\left(-\dfrac{35}{4}\right)\sqrt{50}+\left(-9\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{29}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{18}\right)-\left(\left(\dfrac{2}{7}\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{53}{3}\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{4}\right)\\
&=&\left(\dfrac{14}{5}-\left(\dfrac{20}{3}+\left(-\dfrac{175}{4}\right)\sqrt{2}+\left(-27\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{87}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{135}{8}\right)\sqrt{2}\right)-\dfrac{4}{7}\right)-\dfrac{106}{3}\right)-\left(\dfrac{100}{3}\right)\\
&=&\left(-\dfrac{1352}{35}+\left(\dfrac{527}{8}\right)\sqrt{2}\right)-\left(\dfrac{100}{3}\right)\\
&=&-\dfrac{1352}{35}+\left(\dfrac{527}{8}\right)\sqrt{2}+-\dfrac{100}{3}\\
&=&-\dfrac{7556}{105}+\left(\dfrac{527}{8}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{14}{5}-\left(\dfrac{20}{3}+\left(-\dfrac{35}{4}\right)\sqrt{50}+\left(-9\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{29}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{18}\right)-\left(\left(\dfrac{2}{7}\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{53}{3}\right)\sqrt{4}\right)\right)\times\left(\left(\dfrac{50}{3}\right)\sqrt{4}\right)\\
&=&\left(\dfrac{14}{5}-\left(\dfrac{20}{3}+\left(-\dfrac{175}{4}\right)\sqrt{2}+\left(-27\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{87}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{135}{8}\right)\sqrt{2}\right)-\dfrac{4}{7}\right)-\dfrac{106}{3}\right)\times\left(\dfrac{100}{3}\right)\\
&=&\left(-\dfrac{1352}{35}+\left(\dfrac{527}{8}\right)\sqrt{2}\right)\left(\dfrac{100}{3}\right)\\
&=&-\dfrac{27040}{21}+\left(\dfrac{13175}{6}\right)\sqrt{2}\\
&=&-\dfrac{27040}{21}+\left(\dfrac{13175}{6}\right)\sqrt{2}\\
\end{eqnarray*}