L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{32}{9}-\left(\left(-\dfrac{81}{8}\right)\sqrt{49}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{63}\right)+\left(\left(\dfrac{28}{9}\right)\sqrt{28}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{28}\right)+\left(\left(-9\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{6}\right)\sqrt{63}\right)\) et \( Y=\left(\dfrac{33}{8}\right)\sqrt{28}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{32}{9}-\left(\left(-\dfrac{81}{8}\right)\sqrt{49}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{63}\right)+\left(\left(\dfrac{28}{9}\right)\sqrt{28}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{28}\right)+\left(\left(-9\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{6}\right)\sqrt{63}\right)\right)+\left(\left(\dfrac{33}{8}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{32}{9}+\dfrac{567}{8}-\left(\left(\dfrac{9}{4}\right)\sqrt{7}\right)+\left(\left(\dfrac{56}{9}\right)\sqrt{7}\right)-\left(\left(31\right)\sqrt{7}\right)+\left(\left(-18\right)\sqrt{7}\right)-\left(\left(-\dfrac{67}{2}\right)\sqrt{7}\right)\right)+\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)\\
&=&-\dfrac{32}{9}+\dfrac{567}{8}-\left(\left(\dfrac{9}{4}\right)\sqrt{7}\right)+\left(\left(\dfrac{56}{9}\right)\sqrt{7}\right)-\left(\left(31\right)\sqrt{7}\right)+\left(\left(-18\right)\sqrt{7}\right)-\left(\left(-\dfrac{67}{2}\right)\sqrt{7}\right)+\left(\dfrac{33}{4}\right)\sqrt{7}\\
&=&\dfrac{4847}{72}+\left(-\dfrac{59}{18}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{32}{9}-\left(\left(-\dfrac{81}{8}\right)\sqrt{49}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{63}\right)+\left(\left(\dfrac{28}{9}\right)\sqrt{28}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{28}\right)+\left(\left(-9\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{6}\right)\sqrt{63}\right)\right)-\left(\left(\dfrac{33}{8}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{32}{9}+\dfrac{567}{8}-\left(\left(\dfrac{9}{4}\right)\sqrt{7}\right)+\left(\left(\dfrac{56}{9}\right)\sqrt{7}\right)-\left(\left(31\right)\sqrt{7}\right)+\left(\left(-18\right)\sqrt{7}\right)-\left(\left(-\dfrac{67}{2}\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{4847}{72}+\left(-\dfrac{415}{36}\right)\sqrt{7}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)\\
&=&\dfrac{4847}{72}+\left(-\dfrac{415}{36}\right)\sqrt{7}+\left(-\dfrac{33}{4}\right)\sqrt{7}\\
&=&\dfrac{4847}{72}+\left(-\dfrac{178}{9}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{32}{9}-\left(\left(-\dfrac{81}{8}\right)\sqrt{49}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{63}\right)+\left(\left(\dfrac{28}{9}\right)\sqrt{28}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{28}\right)+\left(\left(-9\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{6}\right)\sqrt{63}\right)\right)\times\left(\left(\dfrac{33}{8}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{32}{9}+\dfrac{567}{8}-\left(\left(\dfrac{9}{4}\right)\sqrt{7}\right)+\left(\left(\dfrac{56}{9}\right)\sqrt{7}\right)-\left(\left(31\right)\sqrt{7}\right)+\left(\left(-18\right)\sqrt{7}\right)-\left(\left(-\dfrac{67}{2}\right)\sqrt{7}\right)\right)\times\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{4847}{72}+\left(-\dfrac{415}{36}\right)\sqrt{7}\right)\left(\left(\dfrac{33}{4}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{53317}{96}\right)\sqrt{7}+\left(-\dfrac{4565}{48}\right)\sqrt{49}\\
&=&\left(\dfrac{53317}{96}\right)\sqrt{7}-\dfrac{31955}{48}\\
\end{eqnarray*}