L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{31}{8}\right)\sqrt{4}\) et \( Y=\left(\left(\left(-\dfrac{29}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{33}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{18}\right)\right)+1-\left(\left(-\dfrac{73}{8}\right)\sqrt{8}-\dfrac{37}{6}\right)-\left(\left(\left(\dfrac{59}{4}\right)\sqrt{18}\right)-\dfrac{2}{3}-\left(\left(\dfrac{59}{4}\right)\sqrt{18}\right)\right)\) . 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{31}{8}\right)\sqrt{4}\right)+\left(\left(\left(\left(-\dfrac{29}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{33}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{18}\right)\right)+1-\left(\left(-\dfrac{73}{8}\right)\sqrt{8}-\dfrac{37}{6}\right)-\left(\left(\left(\dfrac{59}{4}\right)\sqrt{18}\right)-\dfrac{2}{3}-\left(\left(\dfrac{59}{4}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\dfrac{31}{4}\right)+\left(\left(\left(\left(-29\right)\sqrt{2}\right)+\dfrac{66}{7}-\left(\left(-53\right)\sqrt{2}\right)\right)+1-\left(\left(-\dfrac{73}{4}\right)\sqrt{2}-\dfrac{37}{6}\right)-\left(\left(\left(\dfrac{177}{4}\right)\sqrt{2}\right)-\dfrac{2}{3}-\left(\left(\dfrac{177}{4}\right)\sqrt{2}\right)\right)\right)\\
&=&\dfrac{31}{4}+\left(\left(\left(-29\right)\sqrt{2}\right)+\dfrac{66}{7}-\left(\left(-53\right)\sqrt{2}\right)\right)+1-\left(\left(-\dfrac{73}{4}\right)\sqrt{2}-\dfrac{37}{6}\right)-\left(\left(\left(\dfrac{177}{4}\right)\sqrt{2}\right)-\dfrac{2}{3}-\left(\left(\dfrac{177}{4}\right)\sqrt{2}\right)\right)\\
&=&\dfrac{2101}{84}+\left(\dfrac{169}{4}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{31}{8}\right)\sqrt{4}\right)-\left(\left(\left(\left(-\dfrac{29}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{33}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{18}\right)\right)+1-\left(\left(-\dfrac{73}{8}\right)\sqrt{8}-\dfrac{37}{6}\right)-\left(\left(\left(\dfrac{59}{4}\right)\sqrt{18}\right)-\dfrac{2}{3}-\left(\left(\dfrac{59}{4}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\dfrac{31}{4}\right)-\left(\left(\left(\left(-29\right)\sqrt{2}\right)+\dfrac{66}{7}-\left(\left(-53\right)\sqrt{2}\right)\right)+1-\left(\left(-\dfrac{73}{4}\right)\sqrt{2}-\dfrac{37}{6}\right)-\left(\left(\left(\dfrac{177}{4}\right)\sqrt{2}\right)-\dfrac{2}{3}-\left(\left(\dfrac{177}{4}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\dfrac{31}{4}\right)-\left(\left(\dfrac{169}{4}\right)\sqrt{2}+\dfrac{725}{42}\right)\\
&=&\dfrac{31}{4}+\left(-\dfrac{169}{4}\right)\sqrt{2}-\dfrac{725}{42}\\
&=&-\dfrac{799}{84}+\left(-\dfrac{169}{4}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{31}{8}\right)\sqrt{4}\right)\times\left(\left(\left(\left(-\dfrac{29}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{33}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{18}\right)\right)+1-\left(\left(-\dfrac{73}{8}\right)\sqrt{8}-\dfrac{37}{6}\right)-\left(\left(\left(\dfrac{59}{4}\right)\sqrt{18}\right)-\dfrac{2}{3}-\left(\left(\dfrac{59}{4}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\dfrac{31}{4}\right)\times\left(\left(\left(\left(-29\right)\sqrt{2}\right)+\dfrac{66}{7}-\left(\left(-53\right)\sqrt{2}\right)\right)+1-\left(\left(-\dfrac{73}{4}\right)\sqrt{2}-\dfrac{37}{6}\right)-\left(\left(\left(\dfrac{177}{4}\right)\sqrt{2}\right)-\dfrac{2}{3}-\left(\left(\dfrac{177}{4}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\dfrac{31}{4}\right)\left(\left(\dfrac{169}{4}\right)\sqrt{2}+\dfrac{725}{42}\right)\\
&=&\left(\dfrac{5239}{16}\right)\sqrt{2}+\dfrac{22475}{168}\\
&=&\left(\dfrac{5239}{16}\right)\sqrt{2}+\dfrac{22475}{168}\\
\end{eqnarray*}