L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{4}{9}\right)\sqrt{28}-\dfrac{34}{3}+\left(-3\right)\sqrt{28}+\left(\dfrac{43}{6}\right)\sqrt{28}+\left(\dfrac{10}{7}\right)\sqrt{63}-\dfrac{40}{3}+\left(5\right)\sqrt{63}+\left(\dfrac{57}{8}\right)\sqrt{28}+\left(7\right)\sqrt{175}+\left(\dfrac{43}{6}\right)\sqrt{28}-\dfrac{25}{6}+\left(\dfrac{75}{7}\right)\sqrt{49}\) et \( Y=\left(-\dfrac{2}{9}\right)\sqrt{63}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{4}{9}\right)\sqrt{28}-\dfrac{34}{3}+\left(-3\right)\sqrt{28}+\left(\dfrac{43}{6}\right)\sqrt{28}+\left(\dfrac{10}{7}\right)\sqrt{63}-\dfrac{40}{3}+\left(5\right)\sqrt{63}+\left(\dfrac{57}{8}\right)\sqrt{28}+\left(7\right)\sqrt{175}+\left(\dfrac{43}{6}\right)\sqrt{28}-\dfrac{25}{6}+\left(\dfrac{75}{7}\right)\sqrt{49}\right)+\left(\left(-\dfrac{2}{9}\right)\sqrt{63}\right)\\
&=&\left(\left(-\dfrac{8}{9}\right)\sqrt{7}-\dfrac{34}{3}+\left(-6\right)\sqrt{7}+\left(\dfrac{43}{3}\right)\sqrt{7}+\left(\dfrac{30}{7}\right)\sqrt{7}-\dfrac{40}{3}+\left(15\right)\sqrt{7}+\left(\dfrac{57}{4}\right)\sqrt{7}+\left(35\right)\sqrt{7}+\left(\dfrac{43}{3}\right)\sqrt{7}-\dfrac{25}{6}+75\right)+\left(\left(-\dfrac{2}{3}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{8}{9}\right)\sqrt{7}-\dfrac{34}{3}+\left(-6\right)\sqrt{7}+\left(\dfrac{43}{3}\right)\sqrt{7}+\left(\dfrac{30}{7}\right)\sqrt{7}-\dfrac{40}{3}+\left(15\right)\sqrt{7}+\left(\dfrac{57}{4}\right)\sqrt{7}+\left(35\right)\sqrt{7}+\left(\dfrac{43}{3}\right)\sqrt{7}-\dfrac{25}{6}+75+\left(-\dfrac{2}{3}\right)\sqrt{7}\\
&=&\left(\dfrac{22591}{252}\right)\sqrt{7}+\dfrac{277}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{4}{9}\right)\sqrt{28}-\dfrac{34}{3}+\left(-3\right)\sqrt{28}+\left(\dfrac{43}{6}\right)\sqrt{28}+\left(\dfrac{10}{7}\right)\sqrt{63}-\dfrac{40}{3}+\left(5\right)\sqrt{63}+\left(\dfrac{57}{8}\right)\sqrt{28}+\left(7\right)\sqrt{175}+\left(\dfrac{43}{6}\right)\sqrt{28}-\dfrac{25}{6}+\left(\dfrac{75}{7}\right)\sqrt{49}\right)-\left(\left(-\dfrac{2}{9}\right)\sqrt{63}\right)\\
&=&\left(\left(-\dfrac{8}{9}\right)\sqrt{7}-\dfrac{34}{3}+\left(-6\right)\sqrt{7}+\left(\dfrac{43}{3}\right)\sqrt{7}+\left(\dfrac{30}{7}\right)\sqrt{7}-\dfrac{40}{3}+\left(15\right)\sqrt{7}+\left(\dfrac{57}{4}\right)\sqrt{7}+\left(35\right)\sqrt{7}+\left(\dfrac{43}{3}\right)\sqrt{7}-\dfrac{25}{6}+75\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{22759}{252}\right)\sqrt{7}+\dfrac{277}{6}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{22759}{252}\right)\sqrt{7}+\dfrac{277}{6}+\left(\dfrac{2}{3}\right)\sqrt{7}\\
&=&\left(\dfrac{22927}{252}\right)\sqrt{7}+\dfrac{277}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{4}{9}\right)\sqrt{28}-\dfrac{34}{3}+\left(-3\right)\sqrt{28}+\left(\dfrac{43}{6}\right)\sqrt{28}+\left(\dfrac{10}{7}\right)\sqrt{63}-\dfrac{40}{3}+\left(5\right)\sqrt{63}+\left(\dfrac{57}{8}\right)\sqrt{28}+\left(7\right)\sqrt{175}+\left(\dfrac{43}{6}\right)\sqrt{28}-\dfrac{25}{6}+\left(\dfrac{75}{7}\right)\sqrt{49}\right)\times\left(\left(-\dfrac{2}{9}\right)\sqrt{63}\right)\\
&=&\left(\left(-\dfrac{8}{9}\right)\sqrt{7}-\dfrac{34}{3}+\left(-6\right)\sqrt{7}+\left(\dfrac{43}{3}\right)\sqrt{7}+\left(\dfrac{30}{7}\right)\sqrt{7}-\dfrac{40}{3}+\left(15\right)\sqrt{7}+\left(\dfrac{57}{4}\right)\sqrt{7}+\left(35\right)\sqrt{7}+\left(\dfrac{43}{3}\right)\sqrt{7}-\dfrac{25}{6}+75\right)\times\left(\left(-\dfrac{2}{3}\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{22759}{252}\right)\sqrt{7}+\dfrac{277}{6}\right)\left(\left(-\dfrac{2}{3}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{22759}{378}\right)\sqrt{49}+\left(-\dfrac{277}{9}\right)\sqrt{7}\\
&=&-\dfrac{22759}{54}+\left(-\dfrac{277}{9}\right)\sqrt{7}\\
\end{eqnarray*}