L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{8}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{61}{3}\right)\sqrt{75}\right)+\left(\dfrac{22}{5}\right)\sqrt{75}+\left(-\dfrac{81}{5}\right)\sqrt{9}+\left(\dfrac{8}{7}\right)\sqrt{9}-\dfrac{13}{3}+\left(\dfrac{76}{9}\right)\sqrt{9}\) et \( Y=\left(\left(\dfrac{5}{2}\right)\sqrt{75}+\left(-\dfrac{16}{3}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{37}{3}\right)\sqrt{9}\right)-\dfrac{26}{7}-\left(\left(\dfrac{59}{9}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{9}+\left(-\dfrac{35}{2}\right)\sqrt{75}\right)\) . 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(\dfrac{8}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{61}{3}\right)\sqrt{75}\right)+\left(\dfrac{22}{5}\right)\sqrt{75}+\left(-\dfrac{81}{5}\right)\sqrt{9}+\left(\dfrac{8}{7}\right)\sqrt{9}-\dfrac{13}{3}+\left(\dfrac{76}{9}\right)\sqrt{9}\right)+\left(\left(\left(\dfrac{5}{2}\right)\sqrt{75}+\left(-\dfrac{16}{3}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{37}{3}\right)\sqrt{9}\right)-\dfrac{26}{7}-\left(\left(\dfrac{59}{9}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{9}+\left(-\dfrac{35}{2}\right)\sqrt{75}\right)\right)\\
&=&\left(8-\left(\left(\dfrac{305}{3}\right)\sqrt{3}\right)+\left(22\right)\sqrt{3}-\dfrac{243}{5}+\dfrac{24}{7}-\dfrac{13}{3}+\dfrac{76}{3}\right)+\left(\left(\left(\dfrac{25}{2}\right)\sqrt{3}-16\right)-\left(-37-\dfrac{26}{7}-\dfrac{59}{3}\right)-\left(40+\left(-\dfrac{175}{2}\right)\sqrt{3}\right)\right)\\
&=&8-\left(\left(\dfrac{305}{3}\right)\sqrt{3}\right)+\left(22\right)\sqrt{3}-\dfrac{243}{5}+\dfrac{24}{7}-\dfrac{13}{3}+\dfrac{76}{3}+\left(\left(\dfrac{25}{2}\right)\sqrt{3}-16\right)-\left(-37-\dfrac{26}{7}-\dfrac{59}{3}\right)-\left(40+\left(-\dfrac{175}{2}\right)\sqrt{3}\right)\\
&=&-\dfrac{1238}{105}+\left(\dfrac{61}{3}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{8}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{61}{3}\right)\sqrt{75}\right)+\left(\dfrac{22}{5}\right)\sqrt{75}+\left(-\dfrac{81}{5}\right)\sqrt{9}+\left(\dfrac{8}{7}\right)\sqrt{9}-\dfrac{13}{3}+\left(\dfrac{76}{9}\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{5}{2}\right)\sqrt{75}+\left(-\dfrac{16}{3}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{37}{3}\right)\sqrt{9}\right)-\dfrac{26}{7}-\left(\left(\dfrac{59}{9}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{9}+\left(-\dfrac{35}{2}\right)\sqrt{75}\right)\right)\\
&=&\left(8-\left(\left(\dfrac{305}{3}\right)\sqrt{3}\right)+\left(22\right)\sqrt{3}-\dfrac{243}{5}+\dfrac{24}{7}-\dfrac{13}{3}+\dfrac{76}{3}\right)-\left(\left(\left(\dfrac{25}{2}\right)\sqrt{3}-16\right)-\left(-37-\dfrac{26}{7}-\dfrac{59}{3}\right)-\left(40+\left(-\dfrac{175}{2}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{566}{35}+\left(-\dfrac{239}{3}\right)\sqrt{3}\right)-\left(\left(100\right)\sqrt{3}+\dfrac{92}{21}\right)\\
&=&-\dfrac{566}{35}+\left(-\dfrac{239}{3}\right)\sqrt{3}+\left(-100\right)\sqrt{3}-\dfrac{92}{21}\\
&=&-\dfrac{2158}{105}+\left(-\dfrac{539}{3}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{8}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{61}{3}\right)\sqrt{75}\right)+\left(\dfrac{22}{5}\right)\sqrt{75}+\left(-\dfrac{81}{5}\right)\sqrt{9}+\left(\dfrac{8}{7}\right)\sqrt{9}-\dfrac{13}{3}+\left(\dfrac{76}{9}\right)\sqrt{9}\right)\times\left(\left(\left(\dfrac{5}{2}\right)\sqrt{75}+\left(-\dfrac{16}{3}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{37}{3}\right)\sqrt{9}\right)-\dfrac{26}{7}-\left(\left(\dfrac{59}{9}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{9}+\left(-\dfrac{35}{2}\right)\sqrt{75}\right)\right)\\
&=&\left(8-\left(\left(\dfrac{305}{3}\right)\sqrt{3}\right)+\left(22\right)\sqrt{3}-\dfrac{243}{5}+\dfrac{24}{7}-\dfrac{13}{3}+\dfrac{76}{3}\right)\times\left(\left(\left(\dfrac{25}{2}\right)\sqrt{3}-16\right)-\left(-37-\dfrac{26}{7}-\dfrac{59}{3}\right)-\left(40+\left(-\dfrac{175}{2}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{566}{35}+\left(-\dfrac{239}{3}\right)\sqrt{3}\right)\left(\left(100\right)\sqrt{3}+\dfrac{92}{21}\right)\\
&=&\left(-\dfrac{123868}{63}\right)\sqrt{3}-\dfrac{52072}{735}+\left(-\dfrac{23900}{3}\right)\sqrt{9}\\
&=&\left(-\dfrac{123868}{63}\right)\sqrt{3}-\dfrac{17618572}{735}\\
\end{eqnarray*}