L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{11}{2}\right)\sqrt{28}-\dfrac{5}{2}+\left(-\dfrac{3}{5}\right)\sqrt{175}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{175}+\left(\dfrac{27}{2}\right)\sqrt{175}+\left(\dfrac{79}{6}\right)\sqrt{175}\right)\) et \( Y=\left(\left(\left(-\dfrac{35}{2}\right)\sqrt{175}\right)+\dfrac{35}{6}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{28}+\left(\dfrac{37}{6}\right)\sqrt{175}\right)-\left(\left(\left(-\dfrac{77}{8}\right)\sqrt{63}\right)-\left(\left(8\right)\sqrt{28}\right)-4\right)\) . 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{11}{2}\right)\sqrt{28}-\dfrac{5}{2}+\left(-\dfrac{3}{5}\right)\sqrt{175}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{175}+\left(\dfrac{27}{2}\right)\sqrt{175}+\left(\dfrac{79}{6}\right)\sqrt{175}\right)\right)+\left(\left(\left(\left(-\dfrac{35}{2}\right)\sqrt{175}\right)+\dfrac{35}{6}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{28}+\left(\dfrac{37}{6}\right)\sqrt{175}\right)-\left(\left(\left(-\dfrac{77}{8}\right)\sqrt{63}\right)-\left(\left(8\right)\sqrt{28}\right)-4\right)\right)\\
&=&\left(\left(\left(-11\right)\sqrt{7}-\dfrac{5}{2}+\left(-3\right)\sqrt{7}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{7}+\left(\dfrac{135}{2}\right)\sqrt{7}+\left(\dfrac{395}{6}\right)\sqrt{7}\right)\right)+\left(\left(\left(\left(-\dfrac{175}{2}\right)\sqrt{7}\right)+\dfrac{35}{6}\right)-\left(\left(9\right)\sqrt{7}+\left(\dfrac{185}{6}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{231}{8}\right)\sqrt{7}\right)-\left(\left(16\right)\sqrt{7}\right)-4\right)\right)\\
&=&\left(\left(-11\right)\sqrt{7}-\dfrac{5}{2}+\left(-3\right)\sqrt{7}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{7}+\left(\dfrac{135}{2}\right)\sqrt{7}+\left(\dfrac{395}{6}\right)\sqrt{7}\right)+\left(\left(\left(-\dfrac{175}{2}\right)\sqrt{7}\right)+\dfrac{35}{6}\right)-\left(\left(9\right)\sqrt{7}+\left(\dfrac{185}{6}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{231}{8}\right)\sqrt{7}\right)-\left(\left(16\right)\sqrt{7}\right)-4\right)\\
&=&\left(-\dfrac{6415}{24}\right)\sqrt{7}+\dfrac{22}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{28}-\dfrac{5}{2}+\left(-\dfrac{3}{5}\right)\sqrt{175}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{175}+\left(\dfrac{27}{2}\right)\sqrt{175}+\left(\dfrac{79}{6}\right)\sqrt{175}\right)\right)-\left(\left(\left(\left(-\dfrac{35}{2}\right)\sqrt{175}\right)+\dfrac{35}{6}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{28}+\left(\dfrac{37}{6}\right)\sqrt{175}\right)-\left(\left(\left(-\dfrac{77}{8}\right)\sqrt{63}\right)-\left(\left(8\right)\sqrt{28}\right)-4\right)\right)\\
&=&\left(\left(\left(-11\right)\sqrt{7}-\dfrac{5}{2}+\left(-3\right)\sqrt{7}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{7}+\left(\dfrac{135}{2}\right)\sqrt{7}+\left(\dfrac{395}{6}\right)\sqrt{7}\right)\right)-\left(\left(\left(\left(-\dfrac{175}{2}\right)\sqrt{7}\right)+\dfrac{35}{6}\right)-\left(\left(9\right)\sqrt{7}+\left(\dfrac{185}{6}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{231}{8}\right)\sqrt{7}\right)-\left(\left(16\right)\sqrt{7}\right)-4\right)\right)\\
&=&\left(\left(-\dfrac{1109}{6}\right)\sqrt{7}-\dfrac{5}{2}\right)-\left(\left(-\dfrac{1979}{24}\right)\sqrt{7}+\dfrac{59}{6}\right)\\
&=&\left(-\dfrac{1109}{6}\right)\sqrt{7}-\dfrac{5}{2}+\left(\dfrac{1979}{24}\right)\sqrt{7}-\dfrac{59}{6}\\
&=&\left(-\dfrac{819}{8}\right)\sqrt{7}-\dfrac{37}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{28}-\dfrac{5}{2}+\left(-\dfrac{3}{5}\right)\sqrt{175}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{175}+\left(\dfrac{27}{2}\right)\sqrt{175}+\left(\dfrac{79}{6}\right)\sqrt{175}\right)\right)\times\left(\left(\left(\left(-\dfrac{35}{2}\right)\sqrt{175}\right)+\dfrac{35}{6}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{28}+\left(\dfrac{37}{6}\right)\sqrt{175}\right)-\left(\left(\left(-\dfrac{77}{8}\right)\sqrt{63}\right)-\left(\left(8\right)\sqrt{28}\right)-4\right)\right)\\
&=&\left(\left(\left(-11\right)\sqrt{7}-\dfrac{5}{2}+\left(-3\right)\sqrt{7}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{7}+\left(\dfrac{135}{2}\right)\sqrt{7}+\left(\dfrac{395}{6}\right)\sqrt{7}\right)\right)\times\left(\left(\left(\left(-\dfrac{175}{2}\right)\sqrt{7}\right)+\dfrac{35}{6}\right)-\left(\left(9\right)\sqrt{7}+\left(\dfrac{185}{6}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{231}{8}\right)\sqrt{7}\right)-\left(\left(16\right)\sqrt{7}\right)-4\right)\right)\\
&=&\left(\left(-\dfrac{1109}{6}\right)\sqrt{7}-\dfrac{5}{2}\right)\left(\left(-\dfrac{1979}{24}\right)\sqrt{7}+\dfrac{59}{6}\right)\\
&=&\left(\dfrac{2194711}{144}\right)\sqrt{49}+\left(-\dfrac{232039}{144}\right)\sqrt{7}-\dfrac{295}{12}\\
&=&\dfrac{15359437}{144}+\left(-\dfrac{232039}{144}\right)\sqrt{7}\\
\end{eqnarray*}