L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{49}{2}-\dfrac{5}{2}+\left(-\dfrac{9}{5}\right)\sqrt{175}+\dfrac{39}{2}+\left(-\dfrac{37}{3}\right)\sqrt{28}-\dfrac{19}{3}-\left(\left(-3\right)\sqrt{49}\right)+\left(\left(-\dfrac{7}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{10}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{175}\right)\) et \( Y=\left(\left(\dfrac{11}{5}\right)\sqrt{28}\right)-\left(\left(\dfrac{73}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{49}\right)+\left(-7\right)\sqrt{63}+\left(-\dfrac{75}{7}\right)\sqrt{63}+8+7\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{49}{2}-\dfrac{5}{2}+\left(-\dfrac{9}{5}\right)\sqrt{175}+\dfrac{39}{2}+\left(-\dfrac{37}{3}\right)\sqrt{28}-\dfrac{19}{3}-\left(\left(-3\right)\sqrt{49}\right)+\left(\left(-\dfrac{7}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{10}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{175}\right)\right)+\left(\left(\left(\dfrac{11}{5}\right)\sqrt{28}\right)-\left(\left(\dfrac{73}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{49}\right)+\left(-7\right)\sqrt{63}+\left(-\dfrac{75}{7}\right)\sqrt{63}+8+7\right)\\
&=&\left(-\dfrac{49}{2}-\dfrac{5}{2}+\left(-9\right)\sqrt{7}+\dfrac{39}{2}+\left(-\dfrac{74}{3}\right)\sqrt{7}-\dfrac{19}{3}+21-\dfrac{49}{2}+\dfrac{70}{9}+\dfrac{287}{4}-\left(\left(-\dfrac{160}{3}\right)\sqrt{7}\right)\right)+\left(\left(\left(\dfrac{22}{5}\right)\sqrt{7}\right)-\dfrac{511}{8}-\left(\left(-\dfrac{35}{4}\right)\sqrt{7}\right)-\dfrac{77}{2}+\left(-21\right)\sqrt{7}+\left(-\dfrac{225}{7}\right)\sqrt{7}+8+7\right)\\
&=&-\dfrac{49}{2}-\dfrac{5}{2}+\left(-9\right)\sqrt{7}+\dfrac{39}{2}+\left(-\dfrac{74}{3}\right)\sqrt{7}-\dfrac{19}{3}+21-\dfrac{49}{2}+\dfrac{70}{9}+\dfrac{287}{4}-\left(\left(-\dfrac{160}{3}\right)\sqrt{7}\right)+\left(\left(\dfrac{22}{5}\right)\sqrt{7}\right)-\dfrac{511}{8}-\left(\left(-\dfrac{35}{4}\right)\sqrt{7}\right)-\dfrac{77}{2}+\left(-21\right)\sqrt{7}+\left(-\dfrac{225}{7}\right)\sqrt{7}+8+7\\
&=&-\dfrac{1813}{72}+\left(-\dfrac{8537}{420}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{49}{2}-\dfrac{5}{2}+\left(-\dfrac{9}{5}\right)\sqrt{175}+\dfrac{39}{2}+\left(-\dfrac{37}{3}\right)\sqrt{28}-\dfrac{19}{3}-\left(\left(-3\right)\sqrt{49}\right)+\left(\left(-\dfrac{7}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{10}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{175}\right)\right)-\left(\left(\left(\dfrac{11}{5}\right)\sqrt{28}\right)-\left(\left(\dfrac{73}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{49}\right)+\left(-7\right)\sqrt{63}+\left(-\dfrac{75}{7}\right)\sqrt{63}+8+7\right)\\
&=&\left(-\dfrac{49}{2}-\dfrac{5}{2}+\left(-9\right)\sqrt{7}+\dfrac{39}{2}+\left(-\dfrac{74}{3}\right)\sqrt{7}-\dfrac{19}{3}+21-\dfrac{49}{2}+\dfrac{70}{9}+\dfrac{287}{4}-\left(\left(-\dfrac{160}{3}\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{22}{5}\right)\sqrt{7}\right)-\dfrac{511}{8}-\left(\left(-\dfrac{35}{4}\right)\sqrt{7}\right)-\dfrac{77}{2}+\left(-21\right)\sqrt{7}+\left(-\dfrac{225}{7}\right)\sqrt{7}+8+7\right)\\
&=&\left(\dfrac{2239}{36}+\left(\dfrac{59}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{5599}{140}\right)\sqrt{7}-\dfrac{699}{8}\right)\\
&=&\dfrac{2239}{36}+\left(\dfrac{59}{3}\right)\sqrt{7}+\left(\dfrac{5599}{140}\right)\sqrt{7}+\dfrac{699}{8}\\
&=&\dfrac{10769}{72}+\left(\dfrac{25057}{420}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{49}{2}-\dfrac{5}{2}+\left(-\dfrac{9}{5}\right)\sqrt{175}+\dfrac{39}{2}+\left(-\dfrac{37}{3}\right)\sqrt{28}-\dfrac{19}{3}-\left(\left(-3\right)\sqrt{49}\right)+\left(\left(-\dfrac{7}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{10}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{175}\right)\right)\times\left(\left(\left(\dfrac{11}{5}\right)\sqrt{28}\right)-\left(\left(\dfrac{73}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{49}\right)+\left(-7\right)\sqrt{63}+\left(-\dfrac{75}{7}\right)\sqrt{63}+8+7\right)\\
&=&\left(-\dfrac{49}{2}-\dfrac{5}{2}+\left(-9\right)\sqrt{7}+\dfrac{39}{2}+\left(-\dfrac{74}{3}\right)\sqrt{7}-\dfrac{19}{3}+21-\dfrac{49}{2}+\dfrac{70}{9}+\dfrac{287}{4}-\left(\left(-\dfrac{160}{3}\right)\sqrt{7}\right)\right)\times\left(\left(\left(\dfrac{22}{5}\right)\sqrt{7}\right)-\dfrac{511}{8}-\left(\left(-\dfrac{35}{4}\right)\sqrt{7}\right)-\dfrac{77}{2}+\left(-21\right)\sqrt{7}+\left(-\dfrac{225}{7}\right)\sqrt{7}+8+7\right)\\
&=&\left(\dfrac{2239}{36}+\left(\dfrac{59}{3}\right)\sqrt{7}\right)\left(\left(-\dfrac{5599}{140}\right)\sqrt{7}-\dfrac{699}{8}\right)\\
&=&\left(-\dfrac{21196771}{5040}\right)\sqrt{7}-\dfrac{521687}{96}+\left(-\dfrac{330341}{420}\right)\sqrt{49}\\
&=&\left(-\dfrac{21196771}{5040}\right)\sqrt{7}-\dfrac{5251163}{480}\\
\end{eqnarray*}