L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{1}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{175}\right)-\left(\left(\left(\dfrac{33}{5}\right)\sqrt{63}\right)+\dfrac{25}{9}-\left(\left(-\dfrac{31}{2}\right)\sqrt{175}\right)+\dfrac{13}{8}\right)+\dfrac{64}{3}\) et \( Y=\left(\left(-7\right)\sqrt{63}\right)-\left(\left(\left(9\right)\sqrt{63}\right)-\dfrac{17}{3}-\left(\left(-\dfrac{14}{3}\right)\sqrt{175}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{1}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{175}\right)-\left(\left(\left(\dfrac{33}{5}\right)\sqrt{63}\right)+\dfrac{25}{9}-\left(\left(-\dfrac{31}{2}\right)\sqrt{175}\right)+\dfrac{13}{8}\right)+\dfrac{64}{3}\right)+\left(\left(\left(-7\right)\sqrt{63}\right)-\left(\left(\left(9\right)\sqrt{63}\right)-\dfrac{17}{3}-\left(\left(-\dfrac{14}{3}\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{5}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{130}{7}\right)\sqrt{7}\right)-\left(\left(\left(\dfrac{99}{5}\right)\sqrt{7}\right)+\dfrac{25}{9}-\left(\left(-\dfrac{155}{2}\right)\sqrt{7}\right)+\dfrac{13}{8}\right)+\dfrac{64}{3}\right)+\left(\left(\left(-21\right)\sqrt{7}\right)-\left(\left(\left(27\right)\sqrt{7}\right)-\dfrac{17}{3}-\left(\left(-\dfrac{70}{3}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(\left(-\dfrac{5}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{130}{7}\right)\sqrt{7}\right)-\left(\left(\left(\dfrac{99}{5}\right)\sqrt{7}\right)+\dfrac{25}{9}-\left(\left(-\dfrac{155}{2}\right)\sqrt{7}\right)+\dfrac{13}{8}\right)+\dfrac{64}{3}+\left(\left(-21\right)\sqrt{7}\right)-\left(\left(\left(27\right)\sqrt{7}\right)-\dfrac{17}{3}-\left(\left(-\dfrac{70}{3}\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{31663}{210}\right)\sqrt{7}+\dfrac{1627}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{1}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{175}\right)-\left(\left(\left(\dfrac{33}{5}\right)\sqrt{63}\right)+\dfrac{25}{9}-\left(\left(-\dfrac{31}{2}\right)\sqrt{175}\right)+\dfrac{13}{8}\right)+\dfrac{64}{3}\right)-\left(\left(\left(-7\right)\sqrt{63}\right)-\left(\left(\left(9\right)\sqrt{63}\right)-\dfrac{17}{3}-\left(\left(-\dfrac{14}{3}\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{5}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{130}{7}\right)\sqrt{7}\right)-\left(\left(\left(\dfrac{99}{5}\right)\sqrt{7}\right)+\dfrac{25}{9}-\left(\left(-\dfrac{155}{2}\right)\sqrt{7}\right)+\dfrac{13}{8}\right)+\dfrac{64}{3}\right)-\left(\left(\left(-21\right)\sqrt{7}\right)-\left(\left(\left(27\right)\sqrt{7}\right)-\dfrac{17}{3}-\left(\left(-\dfrac{70}{3}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(\left(-\dfrac{5561}{70}\right)\sqrt{7}+\dfrac{1219}{72}\right)-\left(\left(-\dfrac{214}{3}\right)\sqrt{7}+\dfrac{17}{3}\right)\\
&=&\left(-\dfrac{5561}{70}\right)\sqrt{7}+\dfrac{1219}{72}+\left(\dfrac{214}{3}\right)\sqrt{7}-\dfrac{17}{3}\\
&=&\left(-\dfrac{1703}{210}\right)\sqrt{7}+\dfrac{811}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{1}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{26}{7}\right)\sqrt{175}\right)-\left(\left(\left(\dfrac{33}{5}\right)\sqrt{63}\right)+\dfrac{25}{9}-\left(\left(-\dfrac{31}{2}\right)\sqrt{175}\right)+\dfrac{13}{8}\right)+\dfrac{64}{3}\right)\times\left(\left(\left(-7\right)\sqrt{63}\right)-\left(\left(\left(9\right)\sqrt{63}\right)-\dfrac{17}{3}-\left(\left(-\dfrac{14}{3}\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{5}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{130}{7}\right)\sqrt{7}\right)-\left(\left(\left(\dfrac{99}{5}\right)\sqrt{7}\right)+\dfrac{25}{9}-\left(\left(-\dfrac{155}{2}\right)\sqrt{7}\right)+\dfrac{13}{8}\right)+\dfrac{64}{3}\right)\times\left(\left(\left(-21\right)\sqrt{7}\right)-\left(\left(\left(27\right)\sqrt{7}\right)-\dfrac{17}{3}-\left(\left(-\dfrac{70}{3}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(\left(-\dfrac{5561}{70}\right)\sqrt{7}+\dfrac{1219}{72}\right)\left(\left(-\dfrac{214}{3}\right)\sqrt{7}+\dfrac{17}{3}\right)\\
&=&\left(\dfrac{595027}{105}\right)\sqrt{49}+\left(-\dfrac{6266821}{3780}\right)\sqrt{7}+\dfrac{20723}{216}\\
&=&\dfrac{42945559}{1080}+\left(-\dfrac{6266821}{3780}\right)\sqrt{7}\\
\end{eqnarray*}