L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{3}{2}\right)\sqrt{175}\right)-\left(-\dfrac{20}{3}-\left(\left(\dfrac{26}{3}\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{67}{5}\right)\sqrt{49}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{63}+\left(-6\right)\sqrt{28}+\left(\dfrac{7}{9}\right)\sqrt{175}+\left(\dfrac{23}{3}\right)\sqrt{49}\right)\) et \( Y=\left(\left(-8\right)\sqrt{175}\right)-\left(\left(-\dfrac{73}{8}\right)\sqrt{175}\right)+\left(\left(-\dfrac{5}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)+8-\left(\left(\dfrac{81}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{49}{9}\right)\sqrt{63}\right)\) . 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{3}{2}\right)\sqrt{175}\right)-\left(-\dfrac{20}{3}-\left(\left(\dfrac{26}{3}\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{67}{5}\right)\sqrt{49}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{63}+\left(-6\right)\sqrt{28}+\left(\dfrac{7}{9}\right)\sqrt{175}+\left(\dfrac{23}{3}\right)\sqrt{49}\right)\right)+\left(\left(\left(-8\right)\sqrt{175}\right)-\left(\left(-\dfrac{73}{8}\right)\sqrt{175}\right)+\left(\left(-\dfrac{5}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)+8-\left(\left(\dfrac{81}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{49}{9}\right)\sqrt{63}\right)\right)\\
&=&\left(\left(\left(\dfrac{15}{2}\right)\sqrt{7}\right)-\left(-\dfrac{20}{3}-\left(\left(26\right)\sqrt{7}\right)\right)+\dfrac{469}{5}-\left(\left(4\right)\sqrt{7}+\left(-12\right)\sqrt{7}+\left(\dfrac{35}{9}\right)\sqrt{7}+\dfrac{161}{3}\right)\right)+\left(\left(\left(-40\right)\sqrt{7}\right)-\left(\left(-\dfrac{365}{8}\right)\sqrt{7}\right)-\dfrac{35}{6}-\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)+8-\left(\left(\dfrac{162}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{15}{2}\right)\sqrt{7}\right)-\left(-\dfrac{20}{3}-\left(\left(26\right)\sqrt{7}\right)\right)+\dfrac{469}{5}-\left(\left(4\right)\sqrt{7}+\left(-12\right)\sqrt{7}+\left(\dfrac{35}{9}\right)\sqrt{7}+\dfrac{161}{3}\right)+\left(\left(-40\right)\sqrt{7}\right)-\left(\left(-\dfrac{365}{8}\right)\sqrt{7}\right)-\dfrac{35}{6}-\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)+8-\left(\left(\dfrac{162}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{4261}{360}\right)\sqrt{7}+\dfrac{1469}{30}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{3}{2}\right)\sqrt{175}\right)-\left(-\dfrac{20}{3}-\left(\left(\dfrac{26}{3}\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{67}{5}\right)\sqrt{49}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{63}+\left(-6\right)\sqrt{28}+\left(\dfrac{7}{9}\right)\sqrt{175}+\left(\dfrac{23}{3}\right)\sqrt{49}\right)\right)-\left(\left(\left(-8\right)\sqrt{175}\right)-\left(\left(-\dfrac{73}{8}\right)\sqrt{175}\right)+\left(\left(-\dfrac{5}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)+8-\left(\left(\dfrac{81}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{49}{9}\right)\sqrt{63}\right)\right)\\
&=&\left(\left(\left(\dfrac{15}{2}\right)\sqrt{7}\right)-\left(-\dfrac{20}{3}-\left(\left(26\right)\sqrt{7}\right)\right)+\dfrac{469}{5}-\left(\left(4\right)\sqrt{7}+\left(-12\right)\sqrt{7}+\left(\dfrac{35}{9}\right)\sqrt{7}+\dfrac{161}{3}\right)\right)-\left(\left(\left(-40\right)\sqrt{7}\right)-\left(\left(-\dfrac{365}{8}\right)\sqrt{7}\right)-\dfrac{35}{6}-\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)+8-\left(\left(\dfrac{162}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{677}{18}\right)\sqrt{7}+\dfrac{234}{5}\right)-\left(\left(-\dfrac{1031}{40}\right)\sqrt{7}+\dfrac{13}{6}\right)\\
&=&\left(\dfrac{677}{18}\right)\sqrt{7}+\dfrac{234}{5}+\left(\dfrac{1031}{40}\right)\sqrt{7}-\dfrac{13}{6}\\
&=&\left(\dfrac{22819}{360}\right)\sqrt{7}+\dfrac{1339}{30}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{3}{2}\right)\sqrt{175}\right)-\left(-\dfrac{20}{3}-\left(\left(\dfrac{26}{3}\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{67}{5}\right)\sqrt{49}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{63}+\left(-6\right)\sqrt{28}+\left(\dfrac{7}{9}\right)\sqrt{175}+\left(\dfrac{23}{3}\right)\sqrt{49}\right)\right)\times\left(\left(\left(-8\right)\sqrt{175}\right)-\left(\left(-\dfrac{73}{8}\right)\sqrt{175}\right)+\left(\left(-\dfrac{5}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)+8-\left(\left(\dfrac{81}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{49}{9}\right)\sqrt{63}\right)\right)\\
&=&\left(\left(\left(\dfrac{15}{2}\right)\sqrt{7}\right)-\left(-\dfrac{20}{3}-\left(\left(26\right)\sqrt{7}\right)\right)+\dfrac{469}{5}-\left(\left(4\right)\sqrt{7}+\left(-12\right)\sqrt{7}+\left(\dfrac{35}{9}\right)\sqrt{7}+\dfrac{161}{3}\right)\right)\times\left(\left(\left(-40\right)\sqrt{7}\right)-\left(\left(-\dfrac{365}{8}\right)\sqrt{7}\right)-\dfrac{35}{6}-\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)+8-\left(\left(\dfrac{162}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{677}{18}\right)\sqrt{7}+\dfrac{234}{5}\right)\left(\left(-\dfrac{1031}{40}\right)\sqrt{7}+\dfrac{13}{6}\right)\\
&=&\left(-\dfrac{697987}{720}\right)\sqrt{49}+\left(-\dfrac{759226}{675}\right)\sqrt{7}+\dfrac{507}{5}\\
&=&-\dfrac{4812901}{720}+\left(-\dfrac{759226}{675}\right)\sqrt{7}\\
\end{eqnarray*}