L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{43}{6}+\left(5\right)\sqrt{63}+\left(\left(\dfrac{16}{9}\right)\sqrt{175}\right)-\left(\left(6\right)\sqrt{63}\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{175}\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{63}\right)+\left(\dfrac{44}{9}\right)\sqrt{49}+\left(5\right)\sqrt{63}+\left(\dfrac{39}{2}\right)\sqrt{175}\) et \( Y=\left(\left(-\dfrac{11}{6}\right)\sqrt{49}+\left(2\right)\sqrt{63}+\left(\dfrac{41}{9}\right)\sqrt{63}\right)-\left(\left(8\right)\sqrt{49}\right)-\left(\left(\left(\dfrac{63}{4}\right)\sqrt{63}\right)+8-\left(\left(-\dfrac{19}{8}\right)\sqrt{49}\right)\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{63}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{43}{6}+\left(5\right)\sqrt{63}+\left(\left(\dfrac{16}{9}\right)\sqrt{175}\right)-\left(\left(6\right)\sqrt{63}\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{175}\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{63}\right)+\left(\dfrac{44}{9}\right)\sqrt{49}+\left(5\right)\sqrt{63}+\left(\dfrac{39}{2}\right)\sqrt{175}\right)+\left(\left(\left(-\dfrac{11}{6}\right)\sqrt{49}+\left(2\right)\sqrt{63}+\left(\dfrac{41}{9}\right)\sqrt{63}\right)-\left(\left(8\right)\sqrt{49}\right)-\left(\left(\left(\dfrac{63}{4}\right)\sqrt{63}\right)+8-\left(\left(-\dfrac{19}{8}\right)\sqrt{49}\right)\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{63}\right)\right)\\
&=&\left(-\dfrac{43}{6}+\left(15\right)\sqrt{7}+\left(\left(\dfrac{80}{9}\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\left(\left(\dfrac{85}{6}\right)\sqrt{7}\right)-\left(\left(\dfrac{177}{8}\right)\sqrt{7}\right)+\dfrac{308}{9}+\left(15\right)\sqrt{7}+\left(\dfrac{195}{2}\right)\sqrt{7}\right)+\left(\left(-\dfrac{77}{6}+\left(6\right)\sqrt{7}+\left(\dfrac{41}{3}\right)\sqrt{7}\right)-56-\left(\left(\left(\dfrac{189}{4}\right)\sqrt{7}\right)+8+\dfrac{133}{8}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{7}\right)\right)\\
&=&-\dfrac{43}{6}+\left(15\right)\sqrt{7}+\left(\left(\dfrac{80}{9}\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\left(\left(\dfrac{85}{6}\right)\sqrt{7}\right)-\left(\left(\dfrac{177}{8}\right)\sqrt{7}\right)+\dfrac{308}{9}+\left(15\right)\sqrt{7}+\left(\dfrac{195}{2}\right)\sqrt{7}+\left(-\dfrac{77}{6}+\left(6\right)\sqrt{7}+\left(\dfrac{41}{3}\right)\sqrt{7}\right)-56-\left(\left(\left(\dfrac{189}{4}\right)\sqrt{7}\right)+8+\dfrac{133}{8}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{7}\right)\\
&=&-\dfrac{4781}{72}+\left(\dfrac{1441}{72}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{43}{6}+\left(5\right)\sqrt{63}+\left(\left(\dfrac{16}{9}\right)\sqrt{175}\right)-\left(\left(6\right)\sqrt{63}\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{175}\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{63}\right)+\left(\dfrac{44}{9}\right)\sqrt{49}+\left(5\right)\sqrt{63}+\left(\dfrac{39}{2}\right)\sqrt{175}\right)-\left(\left(\left(-\dfrac{11}{6}\right)\sqrt{49}+\left(2\right)\sqrt{63}+\left(\dfrac{41}{9}\right)\sqrt{63}\right)-\left(\left(8\right)\sqrt{49}\right)-\left(\left(\left(\dfrac{63}{4}\right)\sqrt{63}\right)+8-\left(\left(-\dfrac{19}{8}\right)\sqrt{49}\right)\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{63}\right)\right)\\
&=&\left(-\dfrac{43}{6}+\left(15\right)\sqrt{7}+\left(\left(\dfrac{80}{9}\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\left(\left(\dfrac{85}{6}\right)\sqrt{7}\right)-\left(\left(\dfrac{177}{8}\right)\sqrt{7}\right)+\dfrac{308}{9}+\left(15\right)\sqrt{7}+\left(\dfrac{195}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{77}{6}+\left(6\right)\sqrt{7}+\left(\dfrac{41}{3}\right)\sqrt{7}\right)-56-\left(\left(\left(\dfrac{189}{4}\right)\sqrt{7}\right)+8+\dfrac{133}{8}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{7}\right)\right)\\
&=&\left(\dfrac{487}{18}+\left(\dfrac{5911}{72}\right)\sqrt{7}\right)-\left(-\dfrac{2243}{24}+\left(-\dfrac{745}{12}\right)\sqrt{7}\right)\\
&=&\dfrac{487}{18}+\left(\dfrac{5911}{72}\right)\sqrt{7}+\dfrac{2243}{24}+\left(\dfrac{745}{12}\right)\sqrt{7}\\
&=&\dfrac{8677}{72}+\left(\dfrac{10381}{72}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{43}{6}+\left(5\right)\sqrt{63}+\left(\left(\dfrac{16}{9}\right)\sqrt{175}\right)-\left(\left(6\right)\sqrt{63}\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{175}\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{63}\right)+\left(\dfrac{44}{9}\right)\sqrt{49}+\left(5\right)\sqrt{63}+\left(\dfrac{39}{2}\right)\sqrt{175}\right)\times\left(\left(\left(-\dfrac{11}{6}\right)\sqrt{49}+\left(2\right)\sqrt{63}+\left(\dfrac{41}{9}\right)\sqrt{63}\right)-\left(\left(8\right)\sqrt{49}\right)-\left(\left(\left(\dfrac{63}{4}\right)\sqrt{63}\right)+8-\left(\left(-\dfrac{19}{8}\right)\sqrt{49}\right)\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{63}\right)\right)\\
&=&\left(-\dfrac{43}{6}+\left(15\right)\sqrt{7}+\left(\left(\dfrac{80}{9}\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\left(\left(\dfrac{85}{6}\right)\sqrt{7}\right)-\left(\left(\dfrac{177}{8}\right)\sqrt{7}\right)+\dfrac{308}{9}+\left(15\right)\sqrt{7}+\left(\dfrac{195}{2}\right)\sqrt{7}\right)\times\left(\left(-\dfrac{77}{6}+\left(6\right)\sqrt{7}+\left(\dfrac{41}{3}\right)\sqrt{7}\right)-56-\left(\left(\left(\dfrac{189}{4}\right)\sqrt{7}\right)+8+\dfrac{133}{8}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{7}\right)\right)\\
&=&\left(\dfrac{487}{18}+\left(\dfrac{5911}{72}\right)\sqrt{7}\right)\left(-\dfrac{2243}{24}+\left(-\dfrac{745}{12}\right)\sqrt{7}\right)\\
&=&-\dfrac{1092341}{432}+\left(-\dfrac{3490752888}{373248}\right)\sqrt{7}+\left(-\dfrac{4403695}{864}\right)\sqrt{49}\\
&=&-\dfrac{14260556304}{373248}+\left(-\dfrac{3490752888}{373248}\right)\sqrt{7}\\
\end{eqnarray*}