L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{53}{2}\) et \( Y=\left(\dfrac{5}{9}\right)\sqrt{12}+\left(\dfrac{13}{4}\right)\sqrt{12}-\dfrac{32}{5}+\left(\dfrac{35}{6}\right)\sqrt{75}+\left(\left(-\dfrac{65}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{43}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{75}\right)-\left(\left(-8\right)\sqrt{9}\right)+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(-3\right)\sqrt{12}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{53}{2}\right)+\left(\left(\dfrac{5}{9}\right)\sqrt{12}+\left(\dfrac{13}{4}\right)\sqrt{12}-\dfrac{32}{5}+\left(\dfrac{35}{6}\right)\sqrt{75}+\left(\left(-\dfrac{65}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{43}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{75}\right)-\left(\left(-8\right)\sqrt{9}\right)+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(-3\right)\sqrt{12}\right)\\
&=&\left(-\dfrac{53}{2}\right)+\left(\left(\dfrac{10}{9}\right)\sqrt{3}+\left(\dfrac{13}{2}\right)\sqrt{3}-\dfrac{32}{5}+\left(\dfrac{175}{6}\right)\sqrt{3}+\left(\left(-65\right)\sqrt{3}\right)+\dfrac{129}{2}-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)+24-\dfrac{165}{7}+\left(-6\right)\sqrt{3}\right)\\
&=&-\dfrac{53}{2}+\left(\dfrac{10}{9}\right)\sqrt{3}+\left(\dfrac{13}{2}\right)\sqrt{3}-\dfrac{32}{5}+\left(\dfrac{175}{6}\right)\sqrt{3}+\left(\left(-65\right)\sqrt{3}\right)+\dfrac{129}{2}-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)+24-\dfrac{165}{7}+\left(-6\right)\sqrt{3}\\
&=&\dfrac{1121}{35}+\left(-\dfrac{368}{9}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{53}{2}\right)-\left(\left(\dfrac{5}{9}\right)\sqrt{12}+\left(\dfrac{13}{4}\right)\sqrt{12}-\dfrac{32}{5}+\left(\dfrac{35}{6}\right)\sqrt{75}+\left(\left(-\dfrac{65}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{43}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{75}\right)-\left(\left(-8\right)\sqrt{9}\right)+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(-3\right)\sqrt{12}\right)\\
&=&\left(-\dfrac{53}{2}\right)-\left(\left(\dfrac{10}{9}\right)\sqrt{3}+\left(\dfrac{13}{2}\right)\sqrt{3}-\dfrac{32}{5}+\left(\dfrac{175}{6}\right)\sqrt{3}+\left(\left(-65\right)\sqrt{3}\right)+\dfrac{129}{2}-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)+24-\dfrac{165}{7}+\left(-6\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{53}{2}\right)-\left(\left(-\dfrac{368}{9}\right)\sqrt{3}+\dfrac{4097}{70}\right)\\
&=&-\dfrac{53}{2}+\left(\dfrac{368}{9}\right)\sqrt{3}-\dfrac{4097}{70}\\
&=&-\dfrac{2976}{35}+\left(\dfrac{368}{9}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{53}{2}\right)\times\left(\left(\dfrac{5}{9}\right)\sqrt{12}+\left(\dfrac{13}{4}\right)\sqrt{12}-\dfrac{32}{5}+\left(\dfrac{35}{6}\right)\sqrt{75}+\left(\left(-\dfrac{65}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{43}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{75}\right)-\left(\left(-8\right)\sqrt{9}\right)+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(-3\right)\sqrt{12}\right)\\
&=&\left(-\dfrac{53}{2}\right)\times\left(\left(\dfrac{10}{9}\right)\sqrt{3}+\left(\dfrac{13}{2}\right)\sqrt{3}-\dfrac{32}{5}+\left(\dfrac{175}{6}\right)\sqrt{3}+\left(\left(-65\right)\sqrt{3}\right)+\dfrac{129}{2}-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)+24-\dfrac{165}{7}+\left(-6\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{53}{2}\right)\left(\left(-\dfrac{368}{9}\right)\sqrt{3}+\dfrac{4097}{70}\right)\\
&=&\left(\dfrac{9752}{9}\right)\sqrt{3}-\dfrac{217141}{140}\\
&=&\left(\dfrac{9752}{9}\right)\sqrt{3}-\dfrac{217141}{140}\\
\end{eqnarray*}