L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{77}{8}\right)\sqrt{75}\) et \( Y=\dfrac{47}{2}-\left(\left(-\dfrac{15}{4}\right)\sqrt{9}+\left(-6\right)\sqrt{75}+\left(\dfrac{32}{9}\right)\sqrt{12}\right)-\left(-3+\left(-\dfrac{11}{3}\right)\sqrt{9}+\left(-\dfrac{47}{7}\right)\sqrt{9}+\left(-\dfrac{63}{4}\right)\sqrt{75}\right)+\dfrac{51}{7}\) . 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{77}{8}\right)\sqrt{75}\right)+\left(\dfrac{47}{2}-\left(\left(-\dfrac{15}{4}\right)\sqrt{9}+\left(-6\right)\sqrt{75}+\left(\dfrac{32}{9}\right)\sqrt{12}\right)-\left(-3+\left(-\dfrac{11}{3}\right)\sqrt{9}+\left(-\dfrac{47}{7}\right)\sqrt{9}+\left(-\dfrac{63}{4}\right)\sqrt{75}\right)+\dfrac{51}{7}\right)\\
&=&\left(\left(-\dfrac{385}{8}\right)\sqrt{3}\right)+\left(\dfrac{47}{2}-\left(-\dfrac{45}{4}+\left(-30\right)\sqrt{3}+\left(\dfrac{64}{9}\right)\sqrt{3}\right)-\left(-3-11-\dfrac{141}{7}+\left(-\dfrac{315}{4}\right)\sqrt{3}\right)+\dfrac{51}{7}\right)\\
&=&\left(-\dfrac{385}{8}\right)\sqrt{3}+\dfrac{47}{2}-\left(-\dfrac{45}{4}+\left(-30\right)\sqrt{3}+\left(\dfrac{64}{9}\right)\sqrt{3}\right)-\left(-3-11-\dfrac{141}{7}+\left(-\dfrac{315}{4}\right)\sqrt{3}\right)+\dfrac{51}{7}\\
&=&\left(\dfrac{3853}{72}\right)\sqrt{3}+\dfrac{2133}{28}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{77}{8}\right)\sqrt{75}\right)-\left(\dfrac{47}{2}-\left(\left(-\dfrac{15}{4}\right)\sqrt{9}+\left(-6\right)\sqrt{75}+\left(\dfrac{32}{9}\right)\sqrt{12}\right)-\left(-3+\left(-\dfrac{11}{3}\right)\sqrt{9}+\left(-\dfrac{47}{7}\right)\sqrt{9}+\left(-\dfrac{63}{4}\right)\sqrt{75}\right)+\dfrac{51}{7}\right)\\
&=&\left(\left(-\dfrac{385}{8}\right)\sqrt{3}\right)-\left(\dfrac{47}{2}-\left(-\dfrac{45}{4}+\left(-30\right)\sqrt{3}+\left(\dfrac{64}{9}\right)\sqrt{3}\right)-\left(-3-11-\dfrac{141}{7}+\left(-\dfrac{315}{4}\right)\sqrt{3}\right)+\dfrac{51}{7}\right)\\
&=&\left(\left(-\dfrac{385}{8}\right)\sqrt{3}\right)-\left(\dfrac{2133}{28}+\left(\dfrac{3659}{36}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{385}{8}\right)\sqrt{3}+-\dfrac{2133}{28}+\left(-\dfrac{3659}{36}\right)\sqrt{3}\\
&=&\left(-\dfrac{10783}{72}\right)\sqrt{3}-\dfrac{2133}{28}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{77}{8}\right)\sqrt{75}\right)\times\left(\dfrac{47}{2}-\left(\left(-\dfrac{15}{4}\right)\sqrt{9}+\left(-6\right)\sqrt{75}+\left(\dfrac{32}{9}\right)\sqrt{12}\right)-\left(-3+\left(-\dfrac{11}{3}\right)\sqrt{9}+\left(-\dfrac{47}{7}\right)\sqrt{9}+\left(-\dfrac{63}{4}\right)\sqrt{75}\right)+\dfrac{51}{7}\right)\\
&=&\left(\left(-\dfrac{385}{8}\right)\sqrt{3}\right)\times\left(\dfrac{47}{2}-\left(-\dfrac{45}{4}+\left(-30\right)\sqrt{3}+\left(\dfrac{64}{9}\right)\sqrt{3}\right)-\left(-3-11-\dfrac{141}{7}+\left(-\dfrac{315}{4}\right)\sqrt{3}\right)+\dfrac{51}{7}\right)\\
&=&\left(\left(-\dfrac{385}{8}\right)\sqrt{3}\right)\left(\dfrac{2133}{28}+\left(\dfrac{3659}{36}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{117315}{32}\right)\sqrt{3}+\left(-\dfrac{1408715}{288}\right)\sqrt{9}\\
&=&\left(-\dfrac{117315}{32}\right)\sqrt{3}-\dfrac{1408715}{96}\\
\end{eqnarray*}