L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=-\dfrac{7}{2}+\left(\dfrac{29}{8}\right)\sqrt{125}+\dfrac{10}{7}+\left(\dfrac{20}{3}\right)\sqrt{20}+\left(\dfrac{31}{5}\right)\sqrt{45}-\dfrac{7}{2}\) et \( Y=\dfrac{76}{5}+\dfrac{19}{2}-\dfrac{28}{9}+\left(-7\right)\sqrt{125}+\left(\dfrac{5}{2}\right)\sqrt{45}+\left(-\dfrac{19}{3}\right)\sqrt{25}+\left(\dfrac{73}{3}\right)\sqrt{20}+\left(-\dfrac{71}{9}\right)\sqrt{45}+\left(-\dfrac{59}{9}\right)\sqrt{25}+\left(-\dfrac{12}{7}\right)\sqrt{25}+\left(\dfrac{68}{5}\right)\sqrt{25}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{7}{2}+\left(\dfrac{29}{8}\right)\sqrt{125}+\dfrac{10}{7}+\left(\dfrac{20}{3}\right)\sqrt{20}+\left(\dfrac{31}{5}\right)\sqrt{45}-\dfrac{7}{2}\right)+\left(\dfrac{76}{5}+\dfrac{19}{2}-\dfrac{28}{9}+\left(-7\right)\sqrt{125}+\left(\dfrac{5}{2}\right)\sqrt{45}+\left(-\dfrac{19}{3}\right)\sqrt{25}+\left(\dfrac{73}{3}\right)\sqrt{20}+\left(-\dfrac{71}{9}\right)\sqrt{45}+\left(-\dfrac{59}{9}\right)\sqrt{25}+\left(-\dfrac{12}{7}\right)\sqrt{25}+\left(\dfrac{68}{5}\right)\sqrt{25}\right)\\
&=&\left(-\dfrac{7}{2}+\left(\dfrac{145}{8}\right)\sqrt{5}+\dfrac{10}{7}+\left(\dfrac{40}{3}\right)\sqrt{5}+\left(\dfrac{93}{5}\right)\sqrt{5}-\dfrac{7}{2}\right)+\left(\dfrac{76}{5}+\dfrac{19}{2}-\dfrac{28}{9}+\left(-35\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}-\dfrac{95}{3}+\left(\dfrac{146}{3}\right)\sqrt{5}+\left(-\dfrac{71}{3}\right)\sqrt{5}-\dfrac{295}{9}-\dfrac{60}{7}+68\right)\\
&=&-\dfrac{7}{2}+\left(\dfrac{145}{8}\right)\sqrt{5}+\dfrac{10}{7}+\left(\dfrac{40}{3}\right)\sqrt{5}+\left(\dfrac{93}{5}\right)\sqrt{5}-\dfrac{7}{2}+\dfrac{76}{5}+\dfrac{19}{2}-\dfrac{28}{9}+\left(-35\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}-\dfrac{95}{3}+\left(\dfrac{146}{3}\right)\sqrt{5}+\left(-\dfrac{71}{3}\right)\sqrt{5}-\dfrac{295}{9}-\dfrac{60}{7}+68\\
&=&\dfrac{6931}{630}+\left(\dfrac{5707}{120}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{7}{2}+\left(\dfrac{29}{8}\right)\sqrt{125}+\dfrac{10}{7}+\left(\dfrac{20}{3}\right)\sqrt{20}+\left(\dfrac{31}{5}\right)\sqrt{45}-\dfrac{7}{2}\right)-\left(\dfrac{76}{5}+\dfrac{19}{2}-\dfrac{28}{9}+\left(-7\right)\sqrt{125}+\left(\dfrac{5}{2}\right)\sqrt{45}+\left(-\dfrac{19}{3}\right)\sqrt{25}+\left(\dfrac{73}{3}\right)\sqrt{20}+\left(-\dfrac{71}{9}\right)\sqrt{45}+\left(-\dfrac{59}{9}\right)\sqrt{25}+\left(-\dfrac{12}{7}\right)\sqrt{25}+\left(\dfrac{68}{5}\right)\sqrt{25}\right)\\
&=&\left(-\dfrac{7}{2}+\left(\dfrac{145}{8}\right)\sqrt{5}+\dfrac{10}{7}+\left(\dfrac{40}{3}\right)\sqrt{5}+\left(\dfrac{93}{5}\right)\sqrt{5}-\dfrac{7}{2}\right)-\left(\dfrac{76}{5}+\dfrac{19}{2}-\dfrac{28}{9}+\left(-35\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}-\dfrac{95}{3}+\left(\dfrac{146}{3}\right)\sqrt{5}+\left(-\dfrac{71}{3}\right)\sqrt{5}-\dfrac{295}{9}-\dfrac{60}{7}+68\right)\\
&=&\left(-\dfrac{39}{7}+\left(\dfrac{6007}{120}\right)\sqrt{5}\right)-\left(\dfrac{10441}{630}+\left(-\dfrac{5}{2}\right)\sqrt{5}\right)\\
&=&-\dfrac{39}{7}+\left(\dfrac{6007}{120}\right)\sqrt{5}+-\dfrac{10441}{630}+\left(\dfrac{5}{2}\right)\sqrt{5}\\
&=&-\dfrac{1993}{90}+\left(\dfrac{6307}{120}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{7}{2}+\left(\dfrac{29}{8}\right)\sqrt{125}+\dfrac{10}{7}+\left(\dfrac{20}{3}\right)\sqrt{20}+\left(\dfrac{31}{5}\right)\sqrt{45}-\dfrac{7}{2}\right)\times\left(\dfrac{76}{5}+\dfrac{19}{2}-\dfrac{28}{9}+\left(-7\right)\sqrt{125}+\left(\dfrac{5}{2}\right)\sqrt{45}+\left(-\dfrac{19}{3}\right)\sqrt{25}+\left(\dfrac{73}{3}\right)\sqrt{20}+\left(-\dfrac{71}{9}\right)\sqrt{45}+\left(-\dfrac{59}{9}\right)\sqrt{25}+\left(-\dfrac{12}{7}\right)\sqrt{25}+\left(\dfrac{68}{5}\right)\sqrt{25}\right)\\
&=&\left(-\dfrac{7}{2}+\left(\dfrac{145}{8}\right)\sqrt{5}+\dfrac{10}{7}+\left(\dfrac{40}{3}\right)\sqrt{5}+\left(\dfrac{93}{5}\right)\sqrt{5}-\dfrac{7}{2}\right)\times\left(\dfrac{76}{5}+\dfrac{19}{2}-\dfrac{28}{9}+\left(-35\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}-\dfrac{95}{3}+\left(\dfrac{146}{3}\right)\sqrt{5}+\left(-\dfrac{71}{3}\right)\sqrt{5}-\dfrac{295}{9}-\dfrac{60}{7}+68\right)\\
&=&\left(-\dfrac{39}{7}+\left(\dfrac{6007}{120}\right)\sqrt{5}\right)\left(\dfrac{10441}{630}+\left(-\dfrac{5}{2}\right)\sqrt{5}\right)\\
&=&-\dfrac{135733}{1470}+\left(\dfrac{63772087}{75600}\right)\sqrt{5}+\left(-\dfrac{6007}{48}\right)\sqrt{25}\\
&=&-\dfrac{2814813}{3920}+\left(\dfrac{63772087}{75600}\right)\sqrt{5}\\
\end{eqnarray*}