L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{3}{2}\right)\sqrt{75}+\left(\left(\dfrac{59}{3}\right)\sqrt{9}\right)+5\) et \( Y=\left(\dfrac{14}{3}\right)\sqrt{9}+\left(\left(-\dfrac{41}{4}\right)\sqrt{12}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{12}\right)+\left(\left(\dfrac{79}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{41}{7}\right)\sqrt{75}\right)\) . 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{3}{2}\right)\sqrt{75}+\left(\left(\dfrac{59}{3}\right)\sqrt{9}\right)+5\right)+\left(\left(\dfrac{14}{3}\right)\sqrt{9}+\left(\left(-\dfrac{41}{4}\right)\sqrt{12}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{12}\right)+\left(\left(\dfrac{79}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{41}{7}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\dfrac{15}{2}\right)\sqrt{3}+59+5\right)+\left(14+\left(\left(-\dfrac{41}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{41}{2}\right)\sqrt{3}\right)+\left(\left(\dfrac{237}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{205}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{15}{2}\right)\sqrt{3}+59+5+14+\left(\left(-\dfrac{41}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{41}{2}\right)\sqrt{3}\right)+\left(\left(\dfrac{237}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{205}{7}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{169}{14}\right)\sqrt{3}+78\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{3}{2}\right)\sqrt{75}+\left(\left(\dfrac{59}{3}\right)\sqrt{9}\right)+5\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{9}+\left(\left(-\dfrac{41}{4}\right)\sqrt{12}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{12}\right)+\left(\left(\dfrac{79}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{41}{7}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\dfrac{15}{2}\right)\sqrt{3}+59+5\right)-\left(14+\left(\left(-\dfrac{41}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{41}{2}\right)\sqrt{3}\right)+\left(\left(\dfrac{237}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{205}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\dfrac{15}{2}\right)\sqrt{3}+64\right)-\left(14+\left(\dfrac{32}{7}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{15}{2}\right)\sqrt{3}+64+-14+\left(-\dfrac{32}{7}\right)\sqrt{3}\\
&=&\left(\dfrac{41}{14}\right)\sqrt{3}+50\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{3}{2}\right)\sqrt{75}+\left(\left(\dfrac{59}{3}\right)\sqrt{9}\right)+5\right)\times\left(\left(\dfrac{14}{3}\right)\sqrt{9}+\left(\left(-\dfrac{41}{4}\right)\sqrt{12}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{12}\right)+\left(\left(\dfrac{79}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{41}{7}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\dfrac{15}{2}\right)\sqrt{3}+59+5\right)\times\left(14+\left(\left(-\dfrac{41}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{41}{2}\right)\sqrt{3}\right)+\left(\left(\dfrac{237}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{205}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\dfrac{15}{2}\right)\sqrt{3}+64\right)\left(14+\left(\dfrac{32}{7}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{2783}{7}\right)\sqrt{3}+\left(\dfrac{240}{7}\right)\sqrt{9}+896\\
&=&\left(\dfrac{2783}{7}\right)\sqrt{3}+\dfrac{6992}{7}\\
\end{eqnarray*}