L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{23}{4}\right)\sqrt{63}\right)-\dfrac{7}{9}-\left(\left(\dfrac{25}{4}\right)\sqrt{63}\right)+\left(\left(\dfrac{29}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{175}\right)\) et \( Y=\left(-4\right)\sqrt{28}+\left(\left(-4\right)\sqrt{28}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{49}\right)-\dfrac{12}{7}\) . 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(-\dfrac{23}{4}\right)\sqrt{63}\right)-\dfrac{7}{9}-\left(\left(\dfrac{25}{4}\right)\sqrt{63}\right)+\left(\left(\dfrac{29}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{175}\right)\right)+\left(\left(-4\right)\sqrt{28}+\left(\left(-4\right)\sqrt{28}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{49}\right)-\dfrac{12}{7}\right)\\
&=&\left(\left(\left(-\dfrac{69}{4}\right)\sqrt{7}\right)-\dfrac{7}{9}-\left(\left(\dfrac{75}{4}\right)\sqrt{7}\right)+\left(\left(\dfrac{29}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{135}{2}\right)\sqrt{7}\right)\right)+\left(\left(-8\right)\sqrt{7}+\left(\left(-8\right)\sqrt{7}\right)-\dfrac{7}{2}-\dfrac{12}{7}\right)\\
&=&\left(\left(-\dfrac{69}{4}\right)\sqrt{7}\right)-\dfrac{7}{9}-\left(\left(\dfrac{75}{4}\right)\sqrt{7}\right)+\left(\left(\dfrac{29}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{135}{2}\right)\sqrt{7}\right)+\left(-8\right)\sqrt{7}+\left(\left(-8\right)\sqrt{7}\right)-\dfrac{7}{2}-\dfrac{12}{7}\\
&=&\left(\dfrac{151}{6}\right)\sqrt{7}-\dfrac{755}{126}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{23}{4}\right)\sqrt{63}\right)-\dfrac{7}{9}-\left(\left(\dfrac{25}{4}\right)\sqrt{63}\right)+\left(\left(\dfrac{29}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{175}\right)\right)-\left(\left(-4\right)\sqrt{28}+\left(\left(-4\right)\sqrt{28}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{49}\right)-\dfrac{12}{7}\right)\\
&=&\left(\left(\left(-\dfrac{69}{4}\right)\sqrt{7}\right)-\dfrac{7}{9}-\left(\left(\dfrac{75}{4}\right)\sqrt{7}\right)+\left(\left(\dfrac{29}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{135}{2}\right)\sqrt{7}\right)\right)-\left(\left(-8\right)\sqrt{7}+\left(\left(-8\right)\sqrt{7}\right)-\dfrac{7}{2}-\dfrac{12}{7}\right)\\
&=&\left(\left(\dfrac{247}{6}\right)\sqrt{7}-\dfrac{7}{9}\right)-\left(\left(-16\right)\sqrt{7}-\dfrac{73}{14}\right)\\
&=&\left(\dfrac{247}{6}\right)\sqrt{7}-\dfrac{7}{9}+\left(16\right)\sqrt{7}+\dfrac{73}{14}\\
&=&\left(\dfrac{343}{6}\right)\sqrt{7}+\dfrac{559}{126}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{23}{4}\right)\sqrt{63}\right)-\dfrac{7}{9}-\left(\left(\dfrac{25}{4}\right)\sqrt{63}\right)+\left(\left(\dfrac{29}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{175}\right)\right)\times\left(\left(-4\right)\sqrt{28}+\left(\left(-4\right)\sqrt{28}\right)-\left(\left(\dfrac{1}{2}\right)\sqrt{49}\right)-\dfrac{12}{7}\right)\\
&=&\left(\left(\left(-\dfrac{69}{4}\right)\sqrt{7}\right)-\dfrac{7}{9}-\left(\left(\dfrac{75}{4}\right)\sqrt{7}\right)+\left(\left(\dfrac{29}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{135}{2}\right)\sqrt{7}\right)\right)\times\left(\left(-8\right)\sqrt{7}+\left(\left(-8\right)\sqrt{7}\right)-\dfrac{7}{2}-\dfrac{12}{7}\right)\\
&=&\left(\left(\dfrac{247}{6}\right)\sqrt{7}-\dfrac{7}{9}\right)\left(\left(-16\right)\sqrt{7}-\dfrac{73}{14}\right)\\
&=&\left(-\dfrac{1976}{3}\right)\sqrt{49}+\left(-\dfrac{50957}{252}\right)\sqrt{7}+\dfrac{73}{18}\\
&=&-\dfrac{82919}{18}+\left(-\dfrac{50957}{252}\right)\sqrt{7}\\
\end{eqnarray*}