L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{3}{2}\right)\sqrt{9}+\left(6\right)\sqrt{75}+\left(-\dfrac{17}{4}\right)\sqrt{12}+\left(\dfrac{71}{5}\right)\sqrt{9}+\left(\dfrac{15}{2}\right)\sqrt{9}+\left(\left(\dfrac{68}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{49}{2}\right)\sqrt{27}\right)\) et \( Y=\left(-\dfrac{32}{5}\right)\sqrt{27}-9\) . 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{9}+\left(6\right)\sqrt{75}+\left(-\dfrac{17}{4}\right)\sqrt{12}+\left(\dfrac{71}{5}\right)\sqrt{9}+\left(\dfrac{15}{2}\right)\sqrt{9}+\left(\left(\dfrac{68}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{49}{2}\right)\sqrt{27}\right)\right)+\left(\left(-\dfrac{32}{5}\right)\sqrt{27}-9\right)\\
&=&\left(\dfrac{9}{2}+\left(30\right)\sqrt{3}+\left(-\dfrac{17}{2}\right)\sqrt{3}+\dfrac{213}{5}+\dfrac{45}{2}+\left(\left(\dfrac{136}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{147}{2}\right)\sqrt{3}\right)\right)+\left(\left(-\dfrac{96}{5}\right)\sqrt{3}-9\right)\\
&=&\dfrac{9}{2}+\left(30\right)\sqrt{3}+\left(-\dfrac{17}{2}\right)\sqrt{3}+\dfrac{213}{5}+\dfrac{45}{2}+\left(\left(\dfrac{136}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{147}{2}\right)\sqrt{3}\right)+\left(-\dfrac{96}{5}\right)\sqrt{3}-9\\
&=&\dfrac{303}{5}+\left(\dfrac{4251}{70}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{3}{2}\right)\sqrt{9}+\left(6\right)\sqrt{75}+\left(-\dfrac{17}{4}\right)\sqrt{12}+\left(\dfrac{71}{5}\right)\sqrt{9}+\left(\dfrac{15}{2}\right)\sqrt{9}+\left(\left(\dfrac{68}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{49}{2}\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{32}{5}\right)\sqrt{27}-9\right)\\
&=&\left(\dfrac{9}{2}+\left(30\right)\sqrt{3}+\left(-\dfrac{17}{2}\right)\sqrt{3}+\dfrac{213}{5}+\dfrac{45}{2}+\left(\left(\dfrac{136}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{147}{2}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{96}{5}\right)\sqrt{3}-9\right)\\
&=&\left(\dfrac{348}{5}+\left(\dfrac{1119}{14}\right)\sqrt{3}\right)-\left(\left(-\dfrac{96}{5}\right)\sqrt{3}-9\right)\\
&=&\dfrac{348}{5}+\left(\dfrac{1119}{14}\right)\sqrt{3}+\left(\dfrac{96}{5}\right)\sqrt{3}+9\\
&=&\dfrac{393}{5}+\left(\dfrac{6939}{70}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{3}{2}\right)\sqrt{9}+\left(6\right)\sqrt{75}+\left(-\dfrac{17}{4}\right)\sqrt{12}+\left(\dfrac{71}{5}\right)\sqrt{9}+\left(\dfrac{15}{2}\right)\sqrt{9}+\left(\left(\dfrac{68}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{49}{2}\right)\sqrt{27}\right)\right)\times\left(\left(-\dfrac{32}{5}\right)\sqrt{27}-9\right)\\
&=&\left(\dfrac{9}{2}+\left(30\right)\sqrt{3}+\left(-\dfrac{17}{2}\right)\sqrt{3}+\dfrac{213}{5}+\dfrac{45}{2}+\left(\left(\dfrac{136}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{147}{2}\right)\sqrt{3}\right)\right)\times\left(\left(-\dfrac{96}{5}\right)\sqrt{3}-9\right)\\
&=&\left(\dfrac{348}{5}+\left(\dfrac{1119}{14}\right)\sqrt{3}\right)\left(\left(-\dfrac{96}{5}\right)\sqrt{3}-9\right)\\
&=&\left(-\dfrac{719487}{350}\right)\sqrt{3}-\dfrac{3132}{5}+\left(-\dfrac{53712}{35}\right)\sqrt{9}\\
&=&\left(-\dfrac{719487}{350}\right)\sqrt{3}-\dfrac{36612}{7}\\
\end{eqnarray*}