L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{17}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{7}{9}\right)\sqrt{175}\right)\right)-\dfrac{30}{7}\) et \( Y=1+\left(-\dfrac{13}{3}\right)\sqrt{175}+\left(-\dfrac{27}{8}\right)\sqrt{28}\) . 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(\left(\dfrac{17}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{7}{9}\right)\sqrt{175}\right)\right)-\dfrac{30}{7}\right)+\left(1+\left(-\dfrac{13}{3}\right)\sqrt{175}+\left(-\dfrac{27}{8}\right)\sqrt{28}\right)\\
&=&\left(\left(\left(\left(\dfrac{85}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{175}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{35}{9}\right)\sqrt{7}\right)\right)-\dfrac{30}{7}\right)+\left(1+\left(-\dfrac{65}{3}\right)\sqrt{7}+\left(-\dfrac{27}{4}\right)\sqrt{7}\right)\\
&=&\left(\left(\left(\dfrac{85}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{175}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{35}{9}\right)\sqrt{7}\right)\right)-\dfrac{30}{7}+1+\left(-\dfrac{65}{3}\right)\sqrt{7}+\left(-\dfrac{27}{4}\right)\sqrt{7}\\
&=&\left(\dfrac{2237}{36}\right)\sqrt{7}-\dfrac{23}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{17}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{7}{9}\right)\sqrt{175}\right)\right)-\dfrac{30}{7}\right)-\left(1+\left(-\dfrac{13}{3}\right)\sqrt{175}+\left(-\dfrac{27}{8}\right)\sqrt{28}\right)\\
&=&\left(\left(\left(\left(\dfrac{85}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{175}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{35}{9}\right)\sqrt{7}\right)\right)-\dfrac{30}{7}\right)-\left(1+\left(-\dfrac{65}{3}\right)\sqrt{7}+\left(-\dfrac{27}{4}\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{815}{9}\right)\sqrt{7}-\dfrac{30}{7}\right)-\left(1+\left(-\dfrac{341}{12}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{815}{9}\right)\sqrt{7}-\dfrac{30}{7}+-1+\left(\dfrac{341}{12}\right)\sqrt{7}\\
&=&\left(\dfrac{4283}{36}\right)\sqrt{7}-\dfrac{37}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{17}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{7}{9}\right)\sqrt{175}\right)\right)-\dfrac{30}{7}\right)\times\left(1+\left(-\dfrac{13}{3}\right)\sqrt{175}+\left(-\dfrac{27}{8}\right)\sqrt{28}\right)\\
&=&\left(\left(\left(\left(\dfrac{85}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{175}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{35}{9}\right)\sqrt{7}\right)\right)-\dfrac{30}{7}\right)\times\left(1+\left(-\dfrac{65}{3}\right)\sqrt{7}+\left(-\dfrac{27}{4}\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{815}{9}\right)\sqrt{7}-\dfrac{30}{7}\right)\left(1+\left(-\dfrac{341}{12}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{26755}{126}\right)\sqrt{7}+\left(-\dfrac{277915}{108}\right)\sqrt{49}-\dfrac{30}{7}\\
&=&\left(\dfrac{26755}{126}\right)\sqrt{7}-\dfrac{13621075}{756}\\
\end{eqnarray*}