L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{15}{4}\) et \( Y=\left(-\dfrac{5}{2}\right)\sqrt{9}+\left(\left(-\dfrac{59}{6}\right)\sqrt{27}\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)+\left(-\dfrac{59}{6}\right)\sqrt{27}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{15}{4}\right)+\left(\left(-\dfrac{5}{2}\right)\sqrt{9}+\left(\left(-\dfrac{59}{6}\right)\sqrt{27}\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)+\left(-\dfrac{59}{6}\right)\sqrt{27}\right)\\
&=&\left(-\dfrac{15}{4}\right)+\left(-\dfrac{15}{2}+\left(\left(-\dfrac{59}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)+\left(-\dfrac{59}{2}\right)\sqrt{3}\right)\\
&=&-\dfrac{15}{4}-\dfrac{15}{2}+\left(\left(-\dfrac{59}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)+\left(-\dfrac{59}{2}\right)\sqrt{3}\\
&=&-\dfrac{45}{4}+\left(-\dfrac{559}{6}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{15}{4}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{9}+\left(\left(-\dfrac{59}{6}\right)\sqrt{27}\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)+\left(-\dfrac{59}{6}\right)\sqrt{27}\right)\\
&=&\left(-\dfrac{15}{4}\right)-\left(-\dfrac{15}{2}+\left(\left(-\dfrac{59}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)+\left(-\dfrac{59}{2}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{15}{4}\right)-\left(-\dfrac{15}{2}+\left(-\dfrac{559}{6}\right)\sqrt{3}\right)\\
&=&-\dfrac{15}{4}+\dfrac{15}{2}+\left(\dfrac{559}{6}\right)\sqrt{3}\\
&=&\dfrac{15}{4}+\left(\dfrac{559}{6}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{15}{4}\right)\times\left(\left(-\dfrac{5}{2}\right)\sqrt{9}+\left(\left(-\dfrac{59}{6}\right)\sqrt{27}\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)+\left(-\dfrac{59}{6}\right)\sqrt{27}\right)\\
&=&\left(-\dfrac{15}{4}\right)\times\left(-\dfrac{15}{2}+\left(\left(-\dfrac{59}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)+\left(-\dfrac{59}{2}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{15}{4}\right)\left(-\dfrac{15}{2}+\left(-\dfrac{559}{6}\right)\sqrt{3}\right)\\
&=&\dfrac{225}{8}+\left(\dfrac{2795}{8}\right)\sqrt{3}\\
&=&\dfrac{225}{8}+\left(\dfrac{2795}{8}\right)\sqrt{3}\\
\end{eqnarray*}