L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{8}{3}\right)\sqrt{27}\) et \( Y=\left(\left(-\dfrac{9}{7}\right)\sqrt{27}-\dfrac{38}{3}+\left(\dfrac{37}{8}\right)\sqrt{27}+\left(-\dfrac{1}{2}\right)\sqrt{9}+\left(-\dfrac{9}{2}\right)\sqrt{75}\right)-\left(-\dfrac{50}{7}-\dfrac{62}{7}-\left(\left(\dfrac{22}{3}\right)\sqrt{12}\right)\right)-\left(\left(\left(0\right)\sqrt{27}\right)-\dfrac{57}{4}-\left(\left(\dfrac{49}{5}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{75}\right)\right)\) . 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{8}{3}\right)\sqrt{27}\right)+\left(\left(\left(-\dfrac{9}{7}\right)\sqrt{27}-\dfrac{38}{3}+\left(\dfrac{37}{8}\right)\sqrt{27}+\left(-\dfrac{1}{2}\right)\sqrt{9}+\left(-\dfrac{9}{2}\right)\sqrt{75}\right)-\left(-\dfrac{50}{7}-\dfrac{62}{7}-\left(\left(\dfrac{22}{3}\right)\sqrt{12}\right)\right)-\left(\left(\left(0\right)\sqrt{27}\right)-\dfrac{57}{4}-\left(\left(\dfrac{49}{5}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{75}\right)\right)\right)\\
&=&\left(\left(-8\right)\sqrt{3}\right)+\left(\left(\left(-\dfrac{27}{7}\right)\sqrt{3}-\dfrac{38}{3}+\left(\dfrac{111}{8}\right)\sqrt{3}-\dfrac{3}{2}+\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(-\dfrac{50}{7}-\dfrac{62}{7}-\left(\left(\dfrac{44}{3}\right)\sqrt{3}\right)\right)-\left(\left(\left(0\right)\sqrt{3}\right)-\dfrac{57}{4}-\left(\left(\dfrac{98}{5}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)\right)\\
&=&\left(-8\right)\sqrt{3}+\left(\left(-\dfrac{27}{7}\right)\sqrt{3}-\dfrac{38}{3}+\left(\dfrac{111}{8}\right)\sqrt{3}-\dfrac{3}{2}+\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(-\dfrac{50}{7}-\dfrac{62}{7}-\left(\left(\dfrac{44}{3}\right)\sqrt{3}\right)\right)-\left(\left(\left(0\right)\sqrt{3}\right)-\dfrac{57}{4}-\left(\left(\dfrac{98}{5}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{11579}{840}\right)\sqrt{3}+\dfrac{193}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{8}{3}\right)\sqrt{27}\right)-\left(\left(\left(-\dfrac{9}{7}\right)\sqrt{27}-\dfrac{38}{3}+\left(\dfrac{37}{8}\right)\sqrt{27}+\left(-\dfrac{1}{2}\right)\sqrt{9}+\left(-\dfrac{9}{2}\right)\sqrt{75}\right)-\left(-\dfrac{50}{7}-\dfrac{62}{7}-\left(\left(\dfrac{22}{3}\right)\sqrt{12}\right)\right)-\left(\left(\left(0\right)\sqrt{27}\right)-\dfrac{57}{4}-\left(\left(\dfrac{49}{5}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{75}\right)\right)\right)\\
&=&\left(\left(-8\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{27}{7}\right)\sqrt{3}-\dfrac{38}{3}+\left(\dfrac{111}{8}\right)\sqrt{3}-\dfrac{3}{2}+\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(-\dfrac{50}{7}-\dfrac{62}{7}-\left(\left(\dfrac{44}{3}\right)\sqrt{3}\right)\right)-\left(\left(\left(0\right)\sqrt{3}\right)-\dfrac{57}{4}-\left(\left(\dfrac{98}{5}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-8\right)\sqrt{3}\right)-\left(\left(\dfrac{18299}{840}\right)\sqrt{3}+\dfrac{193}{12}\right)\\
&=&\left(-8\right)\sqrt{3}+\left(-\dfrac{18299}{840}\right)\sqrt{3}-\dfrac{193}{12}\\
&=&\left(-\dfrac{25019}{840}\right)\sqrt{3}-\dfrac{193}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{8}{3}\right)\sqrt{27}\right)\times\left(\left(\left(-\dfrac{9}{7}\right)\sqrt{27}-\dfrac{38}{3}+\left(\dfrac{37}{8}\right)\sqrt{27}+\left(-\dfrac{1}{2}\right)\sqrt{9}+\left(-\dfrac{9}{2}\right)\sqrt{75}\right)-\left(-\dfrac{50}{7}-\dfrac{62}{7}-\left(\left(\dfrac{22}{3}\right)\sqrt{12}\right)\right)-\left(\left(\left(0\right)\sqrt{27}\right)-\dfrac{57}{4}-\left(\left(\dfrac{49}{5}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{75}\right)\right)\right)\\
&=&\left(\left(-8\right)\sqrt{3}\right)\times\left(\left(\left(-\dfrac{27}{7}\right)\sqrt{3}-\dfrac{38}{3}+\left(\dfrac{111}{8}\right)\sqrt{3}-\dfrac{3}{2}+\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(-\dfrac{50}{7}-\dfrac{62}{7}-\left(\left(\dfrac{44}{3}\right)\sqrt{3}\right)\right)-\left(\left(\left(0\right)\sqrt{3}\right)-\dfrac{57}{4}-\left(\left(\dfrac{98}{5}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-8\right)\sqrt{3}\right)\left(\left(\dfrac{18299}{840}\right)\sqrt{3}+\dfrac{193}{12}\right)\\
&=&\left(-\dfrac{18299}{105}\right)\sqrt{9}+\left(-\dfrac{386}{3}\right)\sqrt{3}\\
&=&-\dfrac{18299}{35}+\left(-\dfrac{386}{3}\right)\sqrt{3}\\
\end{eqnarray*}