L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{41}{4}+\left(-\dfrac{11}{6}\right)\sqrt{25}+\left(-8\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{125}\right)\) et \( Y=\left(\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\dfrac{68}{7}-\left(\left(-\dfrac{9}{8}\right)\sqrt{45}\right)\right)-\left(\left(-6\right)\sqrt{20}\right)-\left(-\dfrac{39}{7}+\left(\dfrac{19}{2}\right)\sqrt{25}+\left(\dfrac{15}{2}\right)\sqrt{125}\right)\) . 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{41}{4}+\left(-\dfrac{11}{6}\right)\sqrt{25}+\left(-8\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{125}\right)\right)+\left(\left(\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\dfrac{68}{7}-\left(\left(-\dfrac{9}{8}\right)\sqrt{45}\right)\right)-\left(\left(-6\right)\sqrt{20}\right)-\left(-\dfrac{39}{7}+\left(\dfrac{19}{2}\right)\sqrt{25}+\left(\dfrac{15}{2}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\dfrac{41}{4}-\dfrac{55}{6}-40+\left(\dfrac{115}{3}\right)\sqrt{5}\right)-\left(\left(-51\right)\sqrt{5}\right)\right)+\left(\left(\left(\left(27\right)\sqrt{5}\right)-\dfrac{68}{7}-\left(\left(-\dfrac{27}{8}\right)\sqrt{5}\right)\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(-\dfrac{39}{7}+\dfrac{95}{2}+\left(\dfrac{75}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{41}{4}-\dfrac{55}{6}-40+\left(\dfrac{115}{3}\right)\sqrt{5}\right)-\left(\left(-51\right)\sqrt{5}\right)+\left(\left(\left(27\right)\sqrt{5}\right)-\dfrac{68}{7}-\left(\left(-\dfrac{27}{8}\right)\sqrt{5}\right)\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(-\dfrac{39}{7}+\dfrac{95}{2}+\left(\dfrac{75}{2}\right)\sqrt{5}\right)\\
&=&-\dfrac{7607}{84}+\left(\dfrac{2261}{24}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{41}{4}+\left(-\dfrac{11}{6}\right)\sqrt{25}+\left(-8\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{125}\right)\right)-\left(\left(\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\dfrac{68}{7}-\left(\left(-\dfrac{9}{8}\right)\sqrt{45}\right)\right)-\left(\left(-6\right)\sqrt{20}\right)-\left(-\dfrac{39}{7}+\left(\dfrac{19}{2}\right)\sqrt{25}+\left(\dfrac{15}{2}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\dfrac{41}{4}-\dfrac{55}{6}-40+\left(\dfrac{115}{3}\right)\sqrt{5}\right)-\left(\left(-51\right)\sqrt{5}\right)\right)-\left(\left(\left(\left(27\right)\sqrt{5}\right)-\dfrac{68}{7}-\left(\left(-\dfrac{27}{8}\right)\sqrt{5}\right)\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(-\dfrac{39}{7}+\dfrac{95}{2}+\left(\dfrac{75}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{467}{12}+\left(\dfrac{268}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{39}{8}\right)\sqrt{5}-\dfrac{723}{14}\right)\\
&=&-\dfrac{467}{12}+\left(\dfrac{268}{3}\right)\sqrt{5}+\left(-\dfrac{39}{8}\right)\sqrt{5}+\dfrac{723}{14}\\
&=&\dfrac{1069}{84}+\left(\dfrac{2027}{24}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{41}{4}+\left(-\dfrac{11}{6}\right)\sqrt{25}+\left(-8\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{125}\right)\right)\times\left(\left(\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\dfrac{68}{7}-\left(\left(-\dfrac{9}{8}\right)\sqrt{45}\right)\right)-\left(\left(-6\right)\sqrt{20}\right)-\left(-\dfrac{39}{7}+\left(\dfrac{19}{2}\right)\sqrt{25}+\left(\dfrac{15}{2}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\dfrac{41}{4}-\dfrac{55}{6}-40+\left(\dfrac{115}{3}\right)\sqrt{5}\right)-\left(\left(-51\right)\sqrt{5}\right)\right)\times\left(\left(\left(\left(27\right)\sqrt{5}\right)-\dfrac{68}{7}-\left(\left(-\dfrac{27}{8}\right)\sqrt{5}\right)\right)-\left(\left(-12\right)\sqrt{5}\right)-\left(-\dfrac{39}{7}+\dfrac{95}{2}+\left(\dfrac{75}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{467}{12}+\left(\dfrac{268}{3}\right)\sqrt{5}\right)\left(\left(\dfrac{39}{8}\right)\sqrt{5}-\dfrac{723}{14}\right)\\
&=&\left(-\dfrac{1075905}{224}\right)\sqrt{5}+\dfrac{112547}{56}+\left(\dfrac{871}{2}\right)\sqrt{25}\\
&=&\left(-\dfrac{1075905}{224}\right)\sqrt{5}+\dfrac{234487}{56}\\
\end{eqnarray*}