L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{78}{5}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{49}\right)-\dfrac{79}{3}\right)-\left(-\dfrac{70}{9}+\left(-\dfrac{9}{2}\right)\sqrt{63}\right)-\left(\left(-5\right)\sqrt{28}+\left(-3\right)\sqrt{28}+\dfrac{47}{9}\right)\) et \( Y=\left(\left(-9\right)\sqrt{175}\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{63}+\left(-4\right)\sqrt{63}\right)-\dfrac{69}{7}\) . 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(\left(\dfrac{78}{5}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{49}\right)-\dfrac{79}{3}\right)-\left(-\dfrac{70}{9}+\left(-\dfrac{9}{2}\right)\sqrt{63}\right)-\left(\left(-5\right)\sqrt{28}+\left(-3\right)\sqrt{28}+\dfrac{47}{9}\right)\right)+\left(\left(\left(-9\right)\sqrt{175}\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{63}+\left(-4\right)\sqrt{63}\right)-\dfrac{69}{7}\right)\\
&=&\left(\left(\left(\left(78\right)\sqrt{7}\right)+56-\dfrac{79}{3}\right)-\left(-\dfrac{70}{9}+\left(-\dfrac{27}{2}\right)\sqrt{7}\right)-\left(\left(-10\right)\sqrt{7}+\left(-6\right)\sqrt{7}+\dfrac{47}{9}\right)\right)+\left(\left(\left(-45\right)\sqrt{7}\right)-\left(\left(-\dfrac{80}{3}\right)\sqrt{7}+\left(-12\right)\sqrt{7}\right)-\dfrac{69}{7}\right)\\
&=&\left(\left(\left(78\right)\sqrt{7}\right)+56-\dfrac{79}{3}\right)-\left(-\dfrac{70}{9}+\left(-\dfrac{27}{2}\right)\sqrt{7}\right)-\left(\left(-10\right)\sqrt{7}+\left(-6\right)\sqrt{7}+\dfrac{47}{9}\right)+\left(\left(-45\right)\sqrt{7}\right)-\left(\left(-\dfrac{80}{3}\right)\sqrt{7}+\left(-12\right)\sqrt{7}\right)-\dfrac{69}{7}\\
&=&\left(\dfrac{607}{6}\right)\sqrt{7}+\dfrac{1409}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{78}{5}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{49}\right)-\dfrac{79}{3}\right)-\left(-\dfrac{70}{9}+\left(-\dfrac{9}{2}\right)\sqrt{63}\right)-\left(\left(-5\right)\sqrt{28}+\left(-3\right)\sqrt{28}+\dfrac{47}{9}\right)\right)-\left(\left(\left(-9\right)\sqrt{175}\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{63}+\left(-4\right)\sqrt{63}\right)-\dfrac{69}{7}\right)\\
&=&\left(\left(\left(\left(78\right)\sqrt{7}\right)+56-\dfrac{79}{3}\right)-\left(-\dfrac{70}{9}+\left(-\dfrac{27}{2}\right)\sqrt{7}\right)-\left(\left(-10\right)\sqrt{7}+\left(-6\right)\sqrt{7}+\dfrac{47}{9}\right)\right)-\left(\left(\left(-45\right)\sqrt{7}\right)-\left(\left(-\dfrac{80}{3}\right)\sqrt{7}+\left(-12\right)\sqrt{7}\right)-\dfrac{69}{7}\right)\\
&=&\left(\left(\dfrac{215}{2}\right)\sqrt{7}+\dfrac{290}{9}\right)-\left(\left(-\dfrac{19}{3}\right)\sqrt{7}-\dfrac{69}{7}\right)\\
&=&\left(\dfrac{215}{2}\right)\sqrt{7}+\dfrac{290}{9}+\left(\dfrac{19}{3}\right)\sqrt{7}+\dfrac{69}{7}\\
&=&\left(\dfrac{683}{6}\right)\sqrt{7}+\dfrac{2651}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{78}{5}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{49}\right)-\dfrac{79}{3}\right)-\left(-\dfrac{70}{9}+\left(-\dfrac{9}{2}\right)\sqrt{63}\right)-\left(\left(-5\right)\sqrt{28}+\left(-3\right)\sqrt{28}+\dfrac{47}{9}\right)\right)\times\left(\left(\left(-9\right)\sqrt{175}\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{63}+\left(-4\right)\sqrt{63}\right)-\dfrac{69}{7}\right)\\
&=&\left(\left(\left(\left(78\right)\sqrt{7}\right)+56-\dfrac{79}{3}\right)-\left(-\dfrac{70}{9}+\left(-\dfrac{27}{2}\right)\sqrt{7}\right)-\left(\left(-10\right)\sqrt{7}+\left(-6\right)\sqrt{7}+\dfrac{47}{9}\right)\right)\times\left(\left(\left(-45\right)\sqrt{7}\right)-\left(\left(-\dfrac{80}{3}\right)\sqrt{7}+\left(-12\right)\sqrt{7}\right)-\dfrac{69}{7}\right)\\
&=&\left(\left(\dfrac{215}{2}\right)\sqrt{7}+\dfrac{290}{9}\right)\left(\left(-\dfrac{19}{3}\right)\sqrt{7}-\dfrac{69}{7}\right)\\
&=&\left(-\dfrac{4085}{6}\right)\sqrt{49}+\left(-\dfrac{477685}{378}\right)\sqrt{7}-\dfrac{6670}{21}\\
&=&-\dfrac{213505}{42}+\left(-\dfrac{477685}{378}\right)\sqrt{7}\\
\end{eqnarray*}