L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{13}{2}\right)\sqrt{49}\right)-\left(\dfrac{16}{9}-\left(\left(-\dfrac{13}{9}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{63}\right)-\left(\left(0\right)\sqrt{28}\right)\right)-\left(\left(\left(\dfrac{37}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{49}\right)-\left(\left(5\right)\sqrt{28}\right)\right)-\left(\dfrac{16}{7}-\dfrac{34}{3}+\left(-1\right)\sqrt{49}+\left(-9\right)\sqrt{28}\right)\) et \( Y=\left(\left(-\dfrac{71}{2}\right)\sqrt{175}+\left(-\dfrac{17}{6}\right)\sqrt{175}+\left(-\dfrac{28}{9}\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{8}\right)\sqrt{63}\right)\) . 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{13}{2}\right)\sqrt{49}\right)-\left(\dfrac{16}{9}-\left(\left(-\dfrac{13}{9}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{63}\right)-\left(\left(0\right)\sqrt{28}\right)\right)-\left(\left(\left(\dfrac{37}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{49}\right)-\left(\left(5\right)\sqrt{28}\right)\right)-\left(\dfrac{16}{7}-\dfrac{34}{3}+\left(-1\right)\sqrt{49}+\left(-9\right)\sqrt{28}\right)\right)+\left(\left(\left(-\dfrac{71}{2}\right)\sqrt{175}+\left(-\dfrac{17}{6}\right)\sqrt{175}+\left(-\dfrac{28}{9}\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{8}\right)\sqrt{63}\right)\right)\\
&=&\left(\dfrac{91}{2}-\left(\dfrac{16}{9}-\left(\left(-\dfrac{13}{3}\right)\sqrt{7}\right)-\left(\left(3\right)\sqrt{7}\right)-\left(\left(0\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{74}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{7}\right)+\dfrac{175}{2}-\left(\left(10\right)\sqrt{7}\right)\right)-\left(\dfrac{16}{7}-\dfrac{34}{3}-7+\left(-18\right)\sqrt{7}\right)\right)+\left(\left(\left(-\dfrac{355}{2}\right)\sqrt{7}+\left(-\dfrac{85}{6}\right)\sqrt{7}+\left(-\dfrac{28}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{117}{8}\right)\sqrt{7}\right)\right)\\
&=&\dfrac{91}{2}-\left(\dfrac{16}{9}-\left(\left(-\dfrac{13}{3}\right)\sqrt{7}\right)-\left(\left(3\right)\sqrt{7}\right)-\left(\left(0\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{74}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{7}\right)+\dfrac{175}{2}-\left(\left(10\right)\sqrt{7}\right)\right)-\left(\dfrac{16}{7}-\dfrac{34}{3}-7+\left(-18\right)\sqrt{7}\right)+\left(\left(-\dfrac{355}{2}\right)\sqrt{7}+\left(-\dfrac{85}{6}\right)\sqrt{7}+\left(-\dfrac{28}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{117}{8}\right)\sqrt{7}\right)\\
&=&-\dfrac{1747}{63}+\left(-\dfrac{1271}{8}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{13}{2}\right)\sqrt{49}\right)-\left(\dfrac{16}{9}-\left(\left(-\dfrac{13}{9}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{63}\right)-\left(\left(0\right)\sqrt{28}\right)\right)-\left(\left(\left(\dfrac{37}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{49}\right)-\left(\left(5\right)\sqrt{28}\right)\right)-\left(\dfrac{16}{7}-\dfrac{34}{3}+\left(-1\right)\sqrt{49}+\left(-9\right)\sqrt{28}\right)\right)-\left(\left(\left(-\dfrac{71}{2}\right)\sqrt{175}+\left(-\dfrac{17}{6}\right)\sqrt{175}+\left(-\dfrac{28}{9}\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{8}\right)\sqrt{63}\right)\right)\\
&=&\left(\dfrac{91}{2}-\left(\dfrac{16}{9}-\left(\left(-\dfrac{13}{3}\right)\sqrt{7}\right)-\left(\left(3\right)\sqrt{7}\right)-\left(\left(0\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{74}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{7}\right)+\dfrac{175}{2}-\left(\left(10\right)\sqrt{7}\right)\right)-\left(\dfrac{16}{7}-\dfrac{34}{3}-7+\left(-18\right)\sqrt{7}\right)\right)-\left(\left(\left(-\dfrac{355}{2}\right)\sqrt{7}+\left(-\dfrac{85}{6}\right)\sqrt{7}+\left(-\dfrac{28}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{117}{8}\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{1747}{63}+\left(\dfrac{55}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{1491}{8}\right)\sqrt{7}\right)\\
&=&-\dfrac{1747}{63}+\left(\dfrac{55}{2}\right)\sqrt{7}+\left(\dfrac{1491}{8}\right)\sqrt{7}\\
&=&-\dfrac{1747}{63}+\left(\dfrac{1711}{8}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{13}{2}\right)\sqrt{49}\right)-\left(\dfrac{16}{9}-\left(\left(-\dfrac{13}{9}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{63}\right)-\left(\left(0\right)\sqrt{28}\right)\right)-\left(\left(\left(\dfrac{37}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{49}\right)-\left(\left(5\right)\sqrt{28}\right)\right)-\left(\dfrac{16}{7}-\dfrac{34}{3}+\left(-1\right)\sqrt{49}+\left(-9\right)\sqrt{28}\right)\right)\times\left(\left(\left(-\dfrac{71}{2}\right)\sqrt{175}+\left(-\dfrac{17}{6}\right)\sqrt{175}+\left(-\dfrac{28}{9}\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{8}\right)\sqrt{63}\right)\right)\\
&=&\left(\dfrac{91}{2}-\left(\dfrac{16}{9}-\left(\left(-\dfrac{13}{3}\right)\sqrt{7}\right)-\left(\left(3\right)\sqrt{7}\right)-\left(\left(0\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{74}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{7}\right)+\dfrac{175}{2}-\left(\left(10\right)\sqrt{7}\right)\right)-\left(\dfrac{16}{7}-\dfrac{34}{3}-7+\left(-18\right)\sqrt{7}\right)\right)\times\left(\left(\left(-\dfrac{355}{2}\right)\sqrt{7}+\left(-\dfrac{85}{6}\right)\sqrt{7}+\left(-\dfrac{28}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{117}{8}\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{1747}{63}+\left(\dfrac{55}{2}\right)\sqrt{7}\right)\left(\left(-\dfrac{1491}{8}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{124037}{24}\right)\sqrt{7}+\left(-\dfrac{82005}{16}\right)\sqrt{49}\\
&=&\left(\dfrac{124037}{24}\right)\sqrt{7}-\dfrac{574035}{16}\\
\end{eqnarray*}