L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{28}{9}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{63}\right)+9\right)-\left(\left(-4\right)\sqrt{49}+\left(\dfrac{58}{5}\right)\sqrt{63}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{175}\right)\) et \( Y=-\dfrac{26}{3}+\dfrac{65}{6}+2-\dfrac{37}{8}+\left(\dfrac{59}{5}\right)\sqrt{63}\) . 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{28}{9}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{63}\right)+9\right)-\left(\left(-4\right)\sqrt{49}+\left(\dfrac{58}{5}\right)\sqrt{63}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{175}\right)\right)+\left(-\dfrac{26}{3}+\dfrac{65}{6}+2-\dfrac{37}{8}+\left(\dfrac{59}{5}\right)\sqrt{63}\right)\\
&=&\left(\left(\dfrac{196}{9}-\left(\left(0\right)\sqrt{7}\right)+9\right)-\left(-28+\left(\dfrac{174}{5}\right)\sqrt{7}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{7}\right)\right)+\left(-\dfrac{26}{3}+\dfrac{65}{6}+2-\dfrac{37}{8}+\left(\dfrac{177}{5}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{196}{9}-\left(\left(0\right)\sqrt{7}\right)+9\right)-\left(-28+\left(\dfrac{174}{5}\right)\sqrt{7}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{7}\right)-\dfrac{26}{3}+\dfrac{65}{6}+2-\dfrac{37}{8}+\left(\dfrac{177}{5}\right)\sqrt{7}\\
&=&\dfrac{4199}{72}+\left(-\dfrac{19}{10}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{28}{9}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{63}\right)+9\right)-\left(\left(-4\right)\sqrt{49}+\left(\dfrac{58}{5}\right)\sqrt{63}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{175}\right)\right)-\left(-\dfrac{26}{3}+\dfrac{65}{6}+2-\dfrac{37}{8}+\left(\dfrac{59}{5}\right)\sqrt{63}\right)\\
&=&\left(\left(\dfrac{196}{9}-\left(\left(0\right)\sqrt{7}\right)+9\right)-\left(-28+\left(\dfrac{174}{5}\right)\sqrt{7}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{7}\right)\right)-\left(-\dfrac{26}{3}+\dfrac{65}{6}+2-\dfrac{37}{8}+\left(\dfrac{177}{5}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{529}{9}+\left(-\dfrac{373}{10}\right)\sqrt{7}\right)-\left(-\dfrac{11}{24}+\left(\dfrac{177}{5}\right)\sqrt{7}\right)\\
&=&\dfrac{529}{9}+\left(-\dfrac{373}{10}\right)\sqrt{7}+\dfrac{11}{24}+\left(-\dfrac{177}{5}\right)\sqrt{7}\\
&=&\dfrac{4265}{72}+\left(-\dfrac{727}{10}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{28}{9}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{63}\right)+9\right)-\left(\left(-4\right)\sqrt{49}+\left(\dfrac{58}{5}\right)\sqrt{63}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{175}\right)\right)\times\left(-\dfrac{26}{3}+\dfrac{65}{6}+2-\dfrac{37}{8}+\left(\dfrac{59}{5}\right)\sqrt{63}\right)\\
&=&\left(\left(\dfrac{196}{9}-\left(\left(0\right)\sqrt{7}\right)+9\right)-\left(-28+\left(\dfrac{174}{5}\right)\sqrt{7}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{7}\right)\right)\times\left(-\dfrac{26}{3}+\dfrac{65}{6}+2-\dfrac{37}{8}+\left(\dfrac{177}{5}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{529}{9}+\left(-\dfrac{373}{10}\right)\sqrt{7}\right)\left(-\dfrac{11}{24}+\left(\dfrac{177}{5}\right)\sqrt{7}\right)\\
&=&-\dfrac{5819}{216}+\left(\dfrac{503479}{240}\right)\sqrt{7}+\left(-\dfrac{66021}{50}\right)\sqrt{49}\\
&=&-\dfrac{50057351}{5400}+\left(\dfrac{503479}{240}\right)\sqrt{7}\\
\end{eqnarray*}