L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{31}{9}\right)\sqrt{27}\) et \( Y=\left(\left(\dfrac{24}{7}\right)\sqrt{9}+\left(-9\right)\sqrt{75}+\left(-7\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{9}\right)-\left(\left(\dfrac{75}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{3}{8}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{9}+\left(-\dfrac{71}{4}\right)\sqrt{9}+\left(\dfrac{64}{9}\right)\sqrt{75}-\dfrac{23}{6}\right)-\left(\left(-\dfrac{47}{6}\right)\sqrt{9}+\left(7\right)\sqrt{75}+\left(-\dfrac{14}{3}\right)\sqrt{75}\right)\) . 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{31}{9}\right)\sqrt{27}\right)+\left(\left(\left(\dfrac{24}{7}\right)\sqrt{9}+\left(-9\right)\sqrt{75}+\left(-7\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{9}\right)-\left(\left(\dfrac{75}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{3}{8}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{9}+\left(-\dfrac{71}{4}\right)\sqrt{9}+\left(\dfrac{64}{9}\right)\sqrt{75}-\dfrac{23}{6}\right)-\left(\left(-\dfrac{47}{6}\right)\sqrt{9}+\left(7\right)\sqrt{75}+\left(-\dfrac{14}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-\dfrac{31}{3}\right)\sqrt{3}\right)+\left(\left(\dfrac{72}{7}+\left(-45\right)\sqrt{3}+\left(-35\right)\sqrt{3}\right)-\left(\left(\left(-28\right)\sqrt{3}\right)-\dfrac{135}{8}-\left(\left(\dfrac{375}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{15}{8}\right)\sqrt{3}\right)\right)-\left(\dfrac{102}{5}-\dfrac{213}{4}+\left(\dfrac{320}{9}\right)\sqrt{3}-\dfrac{23}{6}\right)-\left(-\dfrac{47}{2}+\left(35\right)\sqrt{3}+\left(-\dfrac{70}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{31}{3}\right)\sqrt{3}+\left(\dfrac{72}{7}+\left(-45\right)\sqrt{3}+\left(-35\right)\sqrt{3}\right)-\left(\left(\left(-28\right)\sqrt{3}\right)-\dfrac{135}{8}-\left(\left(\dfrac{375}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{15}{8}\right)\sqrt{3}\right)\right)-\left(\dfrac{102}{5}-\dfrac{213}{4}+\left(\dfrac{320}{9}\right)\sqrt{3}-\dfrac{23}{6}\right)-\left(-\dfrac{47}{2}+\left(35\right)\sqrt{3}+\left(-\dfrac{70}{3}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{1003}{72}\right)\sqrt{3}+\dfrac{73369}{840}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{31}{9}\right)\sqrt{27}\right)-\left(\left(\left(\dfrac{24}{7}\right)\sqrt{9}+\left(-9\right)\sqrt{75}+\left(-7\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{9}\right)-\left(\left(\dfrac{75}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{3}{8}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{9}+\left(-\dfrac{71}{4}\right)\sqrt{9}+\left(\dfrac{64}{9}\right)\sqrt{75}-\dfrac{23}{6}\right)-\left(\left(-\dfrac{47}{6}\right)\sqrt{9}+\left(7\right)\sqrt{75}+\left(-\dfrac{14}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-\dfrac{31}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{72}{7}+\left(-45\right)\sqrt{3}+\left(-35\right)\sqrt{3}\right)-\left(\left(\left(-28\right)\sqrt{3}\right)-\dfrac{135}{8}-\left(\left(\dfrac{375}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{15}{8}\right)\sqrt{3}\right)\right)-\left(\dfrac{102}{5}-\dfrac{213}{4}+\left(\dfrac{320}{9}\right)\sqrt{3}-\dfrac{23}{6}\right)-\left(-\dfrac{47}{2}+\left(35\right)\sqrt{3}+\left(-\dfrac{70}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{31}{3}\right)\sqrt{3}\right)-\left(\dfrac{73369}{840}+\left(-\dfrac{259}{72}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{31}{3}\right)\sqrt{3}+-\dfrac{73369}{840}+\left(\dfrac{259}{72}\right)\sqrt{3}\\
&=&\left(-\dfrac{485}{72}\right)\sqrt{3}-\dfrac{73369}{840}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{31}{9}\right)\sqrt{27}\right)\times\left(\left(\left(\dfrac{24}{7}\right)\sqrt{9}+\left(-9\right)\sqrt{75}+\left(-7\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{9}\right)-\left(\left(\dfrac{75}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{3}{8}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{9}+\left(-\dfrac{71}{4}\right)\sqrt{9}+\left(\dfrac{64}{9}\right)\sqrt{75}-\dfrac{23}{6}\right)-\left(\left(-\dfrac{47}{6}\right)\sqrt{9}+\left(7\right)\sqrt{75}+\left(-\dfrac{14}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-\dfrac{31}{3}\right)\sqrt{3}\right)\times\left(\left(\dfrac{72}{7}+\left(-45\right)\sqrt{3}+\left(-35\right)\sqrt{3}\right)-\left(\left(\left(-28\right)\sqrt{3}\right)-\dfrac{135}{8}-\left(\left(\dfrac{375}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{15}{8}\right)\sqrt{3}\right)\right)-\left(\dfrac{102}{5}-\dfrac{213}{4}+\left(\dfrac{320}{9}\right)\sqrt{3}-\dfrac{23}{6}\right)-\left(-\dfrac{47}{2}+\left(35\right)\sqrt{3}+\left(-\dfrac{70}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{31}{3}\right)\sqrt{3}\right)\left(\dfrac{73369}{840}+\left(-\dfrac{259}{72}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{2274439}{2520}\right)\sqrt{3}+\left(\dfrac{8029}{216}\right)\sqrt{9}\\
&=&\left(-\dfrac{2274439}{2520}\right)\sqrt{3}+\dfrac{8029}{72}\\
\end{eqnarray*}