L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{40}{3}+\left(\dfrac{7}{4}\right)\sqrt{63}+\left(\dfrac{17}{4}\right)\sqrt{175}\right)-\left(\left(\left(0\right)\sqrt{175}\right)-\left(\left(-\dfrac{49}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{63}\right)-\left(\left(-\dfrac{41}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{175}\right)\right)\) et \( Y=\dfrac{35}{3}+\left(-\dfrac{19}{2}\right)\sqrt{175}+\left(-\dfrac{55}{7}\right)\sqrt{28}+\left(-\dfrac{37}{9}\right)\sqrt{175}+\dfrac{35}{3}+\left(-6\right)\sqrt{175}+\left(-\dfrac{57}{8}\right)\sqrt{49}+\dfrac{3}{2}-\dfrac{17}{7}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{40}{3}+\left(\dfrac{7}{4}\right)\sqrt{63}+\left(\dfrac{17}{4}\right)\sqrt{175}\right)-\left(\left(\left(0\right)\sqrt{175}\right)-\left(\left(-\dfrac{49}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{63}\right)-\left(\left(-\dfrac{41}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{175}\right)\right)\right)+\left(\dfrac{35}{3}+\left(-\dfrac{19}{2}\right)\sqrt{175}+\left(-\dfrac{55}{7}\right)\sqrt{28}+\left(-\dfrac{37}{9}\right)\sqrt{175}+\dfrac{35}{3}+\left(-6\right)\sqrt{175}+\left(-\dfrac{57}{8}\right)\sqrt{49}+\dfrac{3}{2}-\dfrac{17}{7}\right)\\
&=&\left(\left(-\dfrac{40}{3}+\left(\dfrac{21}{4}\right)\sqrt{7}+\left(\dfrac{85}{4}\right)\sqrt{7}\right)-\left(\left(\left(0\right)\sqrt{7}\right)+\dfrac{343}{5}-\left(\left(-29\right)\sqrt{7}\right)+\dfrac{287}{8}-\left(\left(-\dfrac{85}{3}\right)\sqrt{7}\right)\right)\right)+\left(\dfrac{35}{3}+\left(-\dfrac{95}{2}\right)\sqrt{7}+\left(-\dfrac{110}{7}\right)\sqrt{7}+\left(-\dfrac{185}{9}\right)\sqrt{7}+\dfrac{35}{3}+\left(-30\right)\sqrt{7}-\dfrac{399}{8}+\dfrac{3}{2}-\dfrac{17}{7}\right)\\
&=&\left(-\dfrac{40}{3}+\left(\dfrac{21}{4}\right)\sqrt{7}+\left(\dfrac{85}{4}\right)\sqrt{7}\right)-\left(\left(\left(0\right)\sqrt{7}\right)+\dfrac{343}{5}-\left(\left(-29\right)\sqrt{7}\right)+\dfrac{287}{8}-\left(\left(-\dfrac{85}{3}\right)\sqrt{7}\right)\right)+\dfrac{35}{3}+\left(-\dfrac{95}{2}\right)\sqrt{7}+\left(-\dfrac{110}{7}\right)\sqrt{7}+\left(-\dfrac{185}{9}\right)\sqrt{7}+\dfrac{35}{3}+\left(-30\right)\sqrt{7}-\dfrac{399}{8}+\dfrac{3}{2}-\dfrac{17}{7}\\
&=&-\dfrac{20339}{140}+\left(-\dfrac{9110}{63}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{40}{3}+\left(\dfrac{7}{4}\right)\sqrt{63}+\left(\dfrac{17}{4}\right)\sqrt{175}\right)-\left(\left(\left(0\right)\sqrt{175}\right)-\left(\left(-\dfrac{49}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{63}\right)-\left(\left(-\dfrac{41}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{175}\right)\right)\right)-\left(\dfrac{35}{3}+\left(-\dfrac{19}{2}\right)\sqrt{175}+\left(-\dfrac{55}{7}\right)\sqrt{28}+\left(-\dfrac{37}{9}\right)\sqrt{175}+\dfrac{35}{3}+\left(-6\right)\sqrt{175}+\left(-\dfrac{57}{8}\right)\sqrt{49}+\dfrac{3}{2}-\dfrac{17}{7}\right)\\
&=&\left(\left(-\dfrac{40}{3}+\left(\dfrac{21}{4}\right)\sqrt{7}+\left(\dfrac{85}{4}\right)\sqrt{7}\right)-\left(\left(\left(0\right)\sqrt{7}\right)+\dfrac{343}{5}-\left(\left(-29\right)\sqrt{7}\right)+\dfrac{287}{8}-\left(\left(-\dfrac{85}{3}\right)\sqrt{7}\right)\right)\right)-\left(\dfrac{35}{3}+\left(-\dfrac{95}{2}\right)\sqrt{7}+\left(-\dfrac{110}{7}\right)\sqrt{7}+\left(-\dfrac{185}{9}\right)\sqrt{7}+\dfrac{35}{3}+\left(-30\right)\sqrt{7}-\dfrac{399}{8}+\dfrac{3}{2}-\dfrac{17}{7}\right)\\
&=&\left(-\dfrac{14137}{120}+\left(-\dfrac{185}{6}\right)\sqrt{7}\right)-\left(-\dfrac{4615}{168}+\left(-\dfrac{14335}{126}\right)\sqrt{7}\right)\\
&=&-\dfrac{14137}{120}+\left(-\dfrac{185}{6}\right)\sqrt{7}+\dfrac{4615}{168}+\left(\dfrac{14335}{126}\right)\sqrt{7}\\
&=&-\dfrac{18971}{210}+\left(\dfrac{5225}{63}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{40}{3}+\left(\dfrac{7}{4}\right)\sqrt{63}+\left(\dfrac{17}{4}\right)\sqrt{175}\right)-\left(\left(\left(0\right)\sqrt{175}\right)-\left(\left(-\dfrac{49}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{63}\right)-\left(\left(-\dfrac{41}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{175}\right)\right)\right)\times\left(\dfrac{35}{3}+\left(-\dfrac{19}{2}\right)\sqrt{175}+\left(-\dfrac{55}{7}\right)\sqrt{28}+\left(-\dfrac{37}{9}\right)\sqrt{175}+\dfrac{35}{3}+\left(-6\right)\sqrt{175}+\left(-\dfrac{57}{8}\right)\sqrt{49}+\dfrac{3}{2}-\dfrac{17}{7}\right)\\
&=&\left(\left(-\dfrac{40}{3}+\left(\dfrac{21}{4}\right)\sqrt{7}+\left(\dfrac{85}{4}\right)\sqrt{7}\right)-\left(\left(\left(0\right)\sqrt{7}\right)+\dfrac{343}{5}-\left(\left(-29\right)\sqrt{7}\right)+\dfrac{287}{8}-\left(\left(-\dfrac{85}{3}\right)\sqrt{7}\right)\right)\right)\times\left(\dfrac{35}{3}+\left(-\dfrac{95}{2}\right)\sqrt{7}+\left(-\dfrac{110}{7}\right)\sqrt{7}+\left(-\dfrac{185}{9}\right)\sqrt{7}+\dfrac{35}{3}+\left(-30\right)\sqrt{7}-\dfrac{399}{8}+\dfrac{3}{2}-\dfrac{17}{7}\right)\\
&=&\left(-\dfrac{14137}{120}+\left(-\dfrac{185}{6}\right)\sqrt{7}\right)\left(-\dfrac{4615}{168}+\left(-\dfrac{14335}{126}\right)\sqrt{7}\right)\\
&=&\dfrac{13048451}{4032}+\left(\dfrac{43436840832}{3048192}\right)\sqrt{7}+\left(\dfrac{2651975}{756}\right)\sqrt{49}\\
&=&\dfrac{12101995908}{435456}+\left(\dfrac{43436840832}{3048192}\right)\sqrt{7}\\
\end{eqnarray*}