L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(1\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{63}\right)-\dfrac{11}{2}+\left(\dfrac{7}{3}\right)\sqrt{63}+\dfrac{49}{3}+\left(-\dfrac{34}{9}\right)\sqrt{63}+\left(-\dfrac{17}{4}\right)\sqrt{175}\) et \( Y=\left(\left(\left(\dfrac{66}{5}\right)\sqrt{63}\right)-0-\left(\left(\dfrac{32}{7}\right)\sqrt{175}\right)\right)+\dfrac{35}{3}\) . 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(1\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{63}\right)-\dfrac{11}{2}+\left(\dfrac{7}{3}\right)\sqrt{63}+\dfrac{49}{3}+\left(-\dfrac{34}{9}\right)\sqrt{63}+\left(-\dfrac{17}{4}\right)\sqrt{175}\right)+\left(\left(\left(\left(\dfrac{66}{5}\right)\sqrt{63}\right)-0-\left(\left(\dfrac{32}{7}\right)\sqrt{175}\right)\right)+\dfrac{35}{3}\right)\\
&=&\left(7-\left(\left(12\right)\sqrt{7}\right)-\dfrac{11}{2}+\left(7\right)\sqrt{7}+\dfrac{49}{3}+\left(-\dfrac{34}{3}\right)\sqrt{7}+\left(-\dfrac{85}{4}\right)\sqrt{7}\right)+\left(\left(\left(\left(\dfrac{198}{5}\right)\sqrt{7}\right)-0-\left(\left(\dfrac{160}{7}\right)\sqrt{7}\right)\right)+\dfrac{35}{3}\right)\\
&=&7-\left(\left(12\right)\sqrt{7}\right)-\dfrac{11}{2}+\left(7\right)\sqrt{7}+\dfrac{49}{3}+\left(-\dfrac{34}{3}\right)\sqrt{7}+\left(-\dfrac{85}{4}\right)\sqrt{7}+\left(\left(\left(\dfrac{198}{5}\right)\sqrt{7}\right)-0-\left(\left(\dfrac{160}{7}\right)\sqrt{7}\right)\right)+\dfrac{35}{3}\\
&=&\dfrac{59}{2}+\left(-\dfrac{8753}{420}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(1\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{63}\right)-\dfrac{11}{2}+\left(\dfrac{7}{3}\right)\sqrt{63}+\dfrac{49}{3}+\left(-\dfrac{34}{9}\right)\sqrt{63}+\left(-\dfrac{17}{4}\right)\sqrt{175}\right)-\left(\left(\left(\left(\dfrac{66}{5}\right)\sqrt{63}\right)-0-\left(\left(\dfrac{32}{7}\right)\sqrt{175}\right)\right)+\dfrac{35}{3}\right)\\
&=&\left(7-\left(\left(12\right)\sqrt{7}\right)-\dfrac{11}{2}+\left(7\right)\sqrt{7}+\dfrac{49}{3}+\left(-\dfrac{34}{3}\right)\sqrt{7}+\left(-\dfrac{85}{4}\right)\sqrt{7}\right)-\left(\left(\left(\left(\dfrac{198}{5}\right)\sqrt{7}\right)-0-\left(\left(\dfrac{160}{7}\right)\sqrt{7}\right)\right)+\dfrac{35}{3}\right)\\
&=&\left(\dfrac{107}{6}+\left(-\dfrac{451}{12}\right)\sqrt{7}\right)-\left(\left(\dfrac{586}{35}\right)\sqrt{7}+\dfrac{35}{3}\right)\\
&=&\dfrac{107}{6}+\left(-\dfrac{451}{12}\right)\sqrt{7}+\left(-\dfrac{586}{35}\right)\sqrt{7}-\dfrac{35}{3}\\
&=&\dfrac{37}{6}+\left(-\dfrac{22817}{420}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(1\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{63}\right)-\dfrac{11}{2}+\left(\dfrac{7}{3}\right)\sqrt{63}+\dfrac{49}{3}+\left(-\dfrac{34}{9}\right)\sqrt{63}+\left(-\dfrac{17}{4}\right)\sqrt{175}\right)\times\left(\left(\left(\left(\dfrac{66}{5}\right)\sqrt{63}\right)-0-\left(\left(\dfrac{32}{7}\right)\sqrt{175}\right)\right)+\dfrac{35}{3}\right)\\
&=&\left(7-\left(\left(12\right)\sqrt{7}\right)-\dfrac{11}{2}+\left(7\right)\sqrt{7}+\dfrac{49}{3}+\left(-\dfrac{34}{3}\right)\sqrt{7}+\left(-\dfrac{85}{4}\right)\sqrt{7}\right)\times\left(\left(\left(\left(\dfrac{198}{5}\right)\sqrt{7}\right)-0-\left(\left(\dfrac{160}{7}\right)\sqrt{7}\right)\right)+\dfrac{35}{3}\right)\\
&=&\left(\dfrac{107}{6}+\left(-\dfrac{451}{12}\right)\sqrt{7}\right)\left(\left(\dfrac{586}{35}\right)\sqrt{7}+\dfrac{35}{3}\right)\\
&=&\left(-\dfrac{176263}{1260}\right)\sqrt{7}+\dfrac{3745}{18}+\left(-\dfrac{132143}{210}\right)\sqrt{49}\\
&=&\left(-\dfrac{176263}{1260}\right)\sqrt{7}-\dfrac{188852}{45}\\
\end{eqnarray*}