L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{45}{2}\right)\sqrt{125}+\left(\dfrac{5}{3}\right)\sqrt{125}-\dfrac{32}{7}+\left(-\dfrac{26}{9}\right)\sqrt{45}+\left(-\dfrac{19}{3}\right)\sqrt{20}+\left(-\dfrac{26}{3}\right)\sqrt{45}+\dfrac{1}{2}-4\) et \( Y=\left(\left(\dfrac{41}{2}\right)\sqrt{20}+\left(-5\right)\sqrt{20}+\left(\dfrac{23}{4}\right)\sqrt{45}\right)-\dfrac{51}{2}-\left(\left(\dfrac{7}{3}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{2}{9}\right)\sqrt{25}\right)-\left(\left(-4\right)\sqrt{25}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{45}\right)\right)-\left(\left(-4\right)\sqrt{125}\right)\) . 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{45}{2}\right)\sqrt{125}+\left(\dfrac{5}{3}\right)\sqrt{125}-\dfrac{32}{7}+\left(-\dfrac{26}{9}\right)\sqrt{45}+\left(-\dfrac{19}{3}\right)\sqrt{20}+\left(-\dfrac{26}{3}\right)\sqrt{45}+\dfrac{1}{2}-4\right)+\left(\left(\left(\dfrac{41}{2}\right)\sqrt{20}+\left(-5\right)\sqrt{20}+\left(\dfrac{23}{4}\right)\sqrt{45}\right)-\dfrac{51}{2}-\left(\left(\dfrac{7}{3}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{2}{9}\right)\sqrt{25}\right)-\left(\left(-4\right)\sqrt{25}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{45}\right)\right)-\left(\left(-4\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\dfrac{225}{2}\right)\sqrt{5}+\left(\dfrac{25}{3}\right)\sqrt{5}-\dfrac{32}{7}+\left(-\dfrac{26}{3}\right)\sqrt{5}+\left(-\dfrac{38}{3}\right)\sqrt{5}+\left(-26\right)\sqrt{5}+\dfrac{1}{2}-4\right)+\left(\left(\left(41\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\left(\dfrac{69}{4}\right)\sqrt{5}\right)-\dfrac{51}{2}-\dfrac{35}{3}-\left(\dfrac{10}{9}+20-\left(\left(\dfrac{75}{2}\right)\sqrt{5}\right)\right)-\left(\left(-20\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{225}{2}\right)\sqrt{5}+\left(\dfrac{25}{3}\right)\sqrt{5}-\dfrac{32}{7}+\left(-\dfrac{26}{3}\right)\sqrt{5}+\left(-\dfrac{38}{3}\right)\sqrt{5}+\left(-26\right)\sqrt{5}+\dfrac{1}{2}-4+\left(\left(41\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\left(\dfrac{69}{4}\right)\sqrt{5}\right)-\dfrac{51}{2}-\dfrac{35}{3}-\left(\dfrac{10}{9}+20-\left(\left(\dfrac{75}{2}\right)\sqrt{5}\right)\right)-\left(\left(-20\right)\sqrt{5}\right)\\
&=&\left(\dfrac{717}{4}\right)\sqrt{5}-\dfrac{4180}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{45}{2}\right)\sqrt{125}+\left(\dfrac{5}{3}\right)\sqrt{125}-\dfrac{32}{7}+\left(-\dfrac{26}{9}\right)\sqrt{45}+\left(-\dfrac{19}{3}\right)\sqrt{20}+\left(-\dfrac{26}{3}\right)\sqrt{45}+\dfrac{1}{2}-4\right)-\left(\left(\left(\dfrac{41}{2}\right)\sqrt{20}+\left(-5\right)\sqrt{20}+\left(\dfrac{23}{4}\right)\sqrt{45}\right)-\dfrac{51}{2}-\left(\left(\dfrac{7}{3}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{2}{9}\right)\sqrt{25}\right)-\left(\left(-4\right)\sqrt{25}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{45}\right)\right)-\left(\left(-4\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\dfrac{225}{2}\right)\sqrt{5}+\left(\dfrac{25}{3}\right)\sqrt{5}-\dfrac{32}{7}+\left(-\dfrac{26}{3}\right)\sqrt{5}+\left(-\dfrac{38}{3}\right)\sqrt{5}+\left(-26\right)\sqrt{5}+\dfrac{1}{2}-4\right)-\left(\left(\left(41\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\left(\dfrac{69}{4}\right)\sqrt{5}\right)-\dfrac{51}{2}-\dfrac{35}{3}-\left(\dfrac{10}{9}+20-\left(\left(\dfrac{75}{2}\right)\sqrt{5}\right)\right)-\left(\left(-20\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{147}{2}\right)\sqrt{5}-\dfrac{113}{14}\right)-\left(\left(\dfrac{423}{4}\right)\sqrt{5}-\dfrac{1049}{18}\right)\\
&=&\left(\dfrac{147}{2}\right)\sqrt{5}-\dfrac{113}{14}+\left(-\dfrac{423}{4}\right)\sqrt{5}+\dfrac{1049}{18}\\
&=&\left(-\dfrac{129}{4}\right)\sqrt{5}+\dfrac{3163}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{45}{2}\right)\sqrt{125}+\left(\dfrac{5}{3}\right)\sqrt{125}-\dfrac{32}{7}+\left(-\dfrac{26}{9}\right)\sqrt{45}+\left(-\dfrac{19}{3}\right)\sqrt{20}+\left(-\dfrac{26}{3}\right)\sqrt{45}+\dfrac{1}{2}-4\right)\times\left(\left(\left(\dfrac{41}{2}\right)\sqrt{20}+\left(-5\right)\sqrt{20}+\left(\dfrac{23}{4}\right)\sqrt{45}\right)-\dfrac{51}{2}-\left(\left(\dfrac{7}{3}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{2}{9}\right)\sqrt{25}\right)-\left(\left(-4\right)\sqrt{25}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{45}\right)\right)-\left(\left(-4\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\dfrac{225}{2}\right)\sqrt{5}+\left(\dfrac{25}{3}\right)\sqrt{5}-\dfrac{32}{7}+\left(-\dfrac{26}{3}\right)\sqrt{5}+\left(-\dfrac{38}{3}\right)\sqrt{5}+\left(-26\right)\sqrt{5}+\dfrac{1}{2}-4\right)\times\left(\left(\left(41\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\left(\dfrac{69}{4}\right)\sqrt{5}\right)-\dfrac{51}{2}-\dfrac{35}{3}-\left(\dfrac{10}{9}+20-\left(\left(\dfrac{75}{2}\right)\sqrt{5}\right)\right)-\left(\left(-20\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{147}{2}\right)\sqrt{5}-\dfrac{113}{14}\right)\left(\left(\dfrac{423}{4}\right)\sqrt{5}-\dfrac{1049}{18}\right)\\
&=&\left(\dfrac{62181}{8}\right)\sqrt{25}+\left(-\dfrac{863011}{168}\right)\sqrt{5}+\dfrac{118537}{252}\\
&=&\dfrac{19824089}{504}+\left(-\dfrac{863011}{168}\right)\sqrt{5}\\
\end{eqnarray*}