L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{61}{4}\right)\sqrt{49}\right)-\left(-5-\left(\left(-\dfrac{14}{5}\right)\sqrt{63}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{175}\right)\right)\) et \( Y=1+\left(\dfrac{5}{4}\right)\sqrt{175}+\left(-\dfrac{13}{8}\right)\sqrt{28}+\left(-\dfrac{43}{9}\right)\sqrt{49}+\left(\dfrac{41}{8}\right)\sqrt{63}+\left(\dfrac{53}{8}\right)\sqrt{49}+\left(\left(-\dfrac{50}{7}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{63}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{49}\right)\) . 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{61}{4}\right)\sqrt{49}\right)-\left(-5-\left(\left(-\dfrac{14}{5}\right)\sqrt{63}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{175}\right)\right)\right)+\left(1+\left(\dfrac{5}{4}\right)\sqrt{175}+\left(-\dfrac{13}{8}\right)\sqrt{28}+\left(-\dfrac{43}{9}\right)\sqrt{49}+\left(\dfrac{41}{8}\right)\sqrt{63}+\left(\dfrac{53}{8}\right)\sqrt{49}+\left(\left(-\dfrac{50}{7}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{63}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{49}\right)\right)\\
&=&\left(\dfrac{427}{4}-\left(-5-\left(\left(-\dfrac{42}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{215}{7}\right)\sqrt{7}\right)\right)\right)+\left(1+\left(\dfrac{25}{4}\right)\sqrt{7}+\left(-\dfrac{13}{4}\right)\sqrt{7}-\dfrac{301}{9}+\left(\dfrac{123}{8}\right)\sqrt{7}+\dfrac{371}{8}-50-\left(\left(0\right)\sqrt{7}\right)+\dfrac{245}{3}-\dfrac{35}{3}\right)\\
&=&\dfrac{427}{4}-\left(-5-\left(\left(-\dfrac{42}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{215}{7}\right)\sqrt{7}\right)\right)+1+\left(\dfrac{25}{4}\right)\sqrt{7}+\left(-\dfrac{13}{4}\right)\sqrt{7}-\dfrac{301}{9}+\left(\dfrac{123}{8}\right)\sqrt{7}+\dfrac{371}{8}-50-\left(\left(0\right)\sqrt{7}\right)+\dfrac{245}{3}-\dfrac{35}{3}\\
&=&\dfrac{10489}{72}+\left(-\dfrac{5807}{280}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{61}{4}\right)\sqrt{49}\right)-\left(-5-\left(\left(-\dfrac{14}{5}\right)\sqrt{63}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{175}\right)\right)\right)-\left(1+\left(\dfrac{5}{4}\right)\sqrt{175}+\left(-\dfrac{13}{8}\right)\sqrt{28}+\left(-\dfrac{43}{9}\right)\sqrt{49}+\left(\dfrac{41}{8}\right)\sqrt{63}+\left(\dfrac{53}{8}\right)\sqrt{49}+\left(\left(-\dfrac{50}{7}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{63}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{49}\right)\right)\\
&=&\left(\dfrac{427}{4}-\left(-5-\left(\left(-\dfrac{42}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{215}{7}\right)\sqrt{7}\right)\right)\right)-\left(1+\left(\dfrac{25}{4}\right)\sqrt{7}+\left(-\dfrac{13}{4}\right)\sqrt{7}-\dfrac{301}{9}+\left(\dfrac{123}{8}\right)\sqrt{7}+\dfrac{371}{8}-50-\left(\left(0\right)\sqrt{7}\right)+\dfrac{245}{3}-\dfrac{35}{3}\right)\\
&=&\left(\dfrac{447}{4}+\left(-\dfrac{1369}{35}\right)\sqrt{7}\right)-\left(\dfrac{2443}{72}+\left(\dfrac{147}{8}\right)\sqrt{7}\right)\\
&=&\dfrac{447}{4}+\left(-\dfrac{1369}{35}\right)\sqrt{7}+-\dfrac{2443}{72}+\left(-\dfrac{147}{8}\right)\sqrt{7}\\
&=&\dfrac{5603}{72}+\left(-\dfrac{16097}{280}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{61}{4}\right)\sqrt{49}\right)-\left(-5-\left(\left(-\dfrac{14}{5}\right)\sqrt{63}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{175}\right)\right)\right)\times\left(1+\left(\dfrac{5}{4}\right)\sqrt{175}+\left(-\dfrac{13}{8}\right)\sqrt{28}+\left(-\dfrac{43}{9}\right)\sqrt{49}+\left(\dfrac{41}{8}\right)\sqrt{63}+\left(\dfrac{53}{8}\right)\sqrt{49}+\left(\left(-\dfrac{50}{7}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{63}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{49}\right)\right)\\
&=&\left(\dfrac{427}{4}-\left(-5-\left(\left(-\dfrac{42}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{215}{7}\right)\sqrt{7}\right)\right)\right)\times\left(1+\left(\dfrac{25}{4}\right)\sqrt{7}+\left(-\dfrac{13}{4}\right)\sqrt{7}-\dfrac{301}{9}+\left(\dfrac{123}{8}\right)\sqrt{7}+\dfrac{371}{8}-50-\left(\left(0\right)\sqrt{7}\right)+\dfrac{245}{3}-\dfrac{35}{3}\right)\\
&=&\left(\dfrac{447}{4}+\left(-\dfrac{1369}{35}\right)\sqrt{7}\right)\left(\dfrac{2443}{72}+\left(\dfrac{147}{8}\right)\sqrt{7}\right)\\
&=&\dfrac{364007}{96}+\left(\dfrac{1045781}{1440}\right)\sqrt{7}+\left(-\dfrac{28749}{40}\right)\sqrt{49}\\
&=&-\dfrac{594881}{480}+\left(\dfrac{1045781}{1440}\right)\sqrt{7}\\
\end{eqnarray*}