L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{5}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{58}{5}\right)\sqrt{20}\right)-\dfrac{4}{3}+\left(\dfrac{61}{4}\right)\sqrt{25}+0+\left(-\dfrac{41}{8}\right)\sqrt{20}+\left(\dfrac{34}{9}\right)\sqrt{20}+\dfrac{19}{7}+0+\left(-\dfrac{15}{4}\right)\sqrt{25}+\left(-7\right)\sqrt{25}\) et \( Y=\left(-\dfrac{38}{5}\right)\sqrt{45}+\left(-\dfrac{23}{3}\right)\sqrt{125}+\left(-7\right)\sqrt{125}+\dfrac{62}{5}-8+\dfrac{29}{8}+\left(\dfrac{19}{3}\right)\sqrt{20}+\left(\dfrac{66}{7}\right)\sqrt{25}+\left(\dfrac{47}{4}\right)\sqrt{125}+\left(\left(\dfrac{41}{8}\right)\sqrt{25}\right)-\left(\left(-\dfrac{17}{7}\right)\sqrt{25}\right)\) . 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{5}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{58}{5}\right)\sqrt{20}\right)-\dfrac{4}{3}+\left(\dfrac{61}{4}\right)\sqrt{25}+0+\left(-\dfrac{41}{8}\right)\sqrt{20}+\left(\dfrac{34}{9}\right)\sqrt{20}+\dfrac{19}{7}+0+\left(-\dfrac{15}{4}\right)\sqrt{25}+\left(-7\right)\sqrt{25}\right)+\left(\left(-\dfrac{38}{5}\right)\sqrt{45}+\left(-\dfrac{23}{3}\right)\sqrt{125}+\left(-7\right)\sqrt{125}+\dfrac{62}{5}-8+\dfrac{29}{8}+\left(\dfrac{19}{3}\right)\sqrt{20}+\left(\dfrac{66}{7}\right)\sqrt{25}+\left(\dfrac{47}{4}\right)\sqrt{125}+\left(\left(\dfrac{41}{8}\right)\sqrt{25}\right)-\left(\left(-\dfrac{17}{7}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{116}{5}\right)\sqrt{5}\right)-\dfrac{4}{3}+\dfrac{305}{4}+0+\left(-\dfrac{41}{4}\right)\sqrt{5}+\left(\dfrac{68}{9}\right)\sqrt{5}+\dfrac{19}{7}+0-\dfrac{75}{4}-35\right)+\left(\left(-\dfrac{114}{5}\right)\sqrt{5}+\left(-\dfrac{115}{3}\right)\sqrt{5}+\left(-35\right)\sqrt{5}+\dfrac{62}{5}-8+\dfrac{29}{8}+\left(\dfrac{38}{3}\right)\sqrt{5}+\dfrac{330}{7}+\left(\dfrac{235}{4}\right)\sqrt{5}+\dfrac{205}{8}+\dfrac{85}{7}\right)\\
&=&\left(\left(-\dfrac{5}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{116}{5}\right)\sqrt{5}\right)-\dfrac{4}{3}+\dfrac{305}{4}+0+\left(-\dfrac{41}{4}\right)\sqrt{5}+\left(\dfrac{68}{9}\right)\sqrt{5}+\dfrac{19}{7}+0-\dfrac{75}{4}-35+\left(-\dfrac{114}{5}\right)\sqrt{5}+\left(-\dfrac{115}{3}\right)\sqrt{5}+\left(-35\right)\sqrt{5}+\dfrac{62}{5}-8+\dfrac{29}{8}+\left(\dfrac{38}{3}\right)\sqrt{5}+\dfrac{330}{7}+\left(\dfrac{235}{4}\right)\sqrt{5}+\dfrac{205}{8}+\dfrac{85}{7}\\
&=&\left(-\dfrac{302}{45}\right)\sqrt{5}+\dfrac{7009}{60}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{5}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{58}{5}\right)\sqrt{20}\right)-\dfrac{4}{3}+\left(\dfrac{61}{4}\right)\sqrt{25}+0+\left(-\dfrac{41}{8}\right)\sqrt{20}+\left(\dfrac{34}{9}\right)\sqrt{20}+\dfrac{19}{7}+0+\left(-\dfrac{15}{4}\right)\sqrt{25}+\left(-7\right)\sqrt{25}\right)-\left(\left(-\dfrac{38}{5}\right)\sqrt{45}+\left(-\dfrac{23}{3}\right)\sqrt{125}+\left(-7\right)\sqrt{125}+\dfrac{62}{5}-8+\dfrac{29}{8}+\left(\dfrac{19}{3}\right)\sqrt{20}+\left(\dfrac{66}{7}\right)\sqrt{25}+\left(\dfrac{47}{4}\right)\sqrt{125}+\left(\left(\dfrac{41}{8}\right)\sqrt{25}\right)-\left(\left(-\dfrac{17}{7}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{116}{5}\right)\sqrt{5}\right)-\dfrac{4}{3}+\dfrac{305}{4}+0+\left(-\dfrac{41}{4}\right)\sqrt{5}+\left(\dfrac{68}{9}\right)\sqrt{5}+\dfrac{19}{7}+0-\dfrac{75}{4}-35\right)-\left(\left(-\dfrac{114}{5}\right)\sqrt{5}+\left(-\dfrac{115}{3}\right)\sqrt{5}+\left(-35\right)\sqrt{5}+\dfrac{62}{5}-8+\dfrac{29}{8}+\left(\dfrac{38}{3}\right)\sqrt{5}+\dfrac{330}{7}+\left(\dfrac{235}{4}\right)\sqrt{5}+\dfrac{205}{8}+\dfrac{85}{7}\right)\\
&=&\left(\left(\dfrac{3241}{180}\right)\sqrt{5}+\dfrac{1003}{42}\right)-\left(\left(-\dfrac{1483}{60}\right)\sqrt{5}+\dfrac{13011}{140}\right)\\
&=&\left(\dfrac{3241}{180}\right)\sqrt{5}+\dfrac{1003}{42}+\left(\dfrac{1483}{60}\right)\sqrt{5}-\dfrac{13011}{140}\\
&=&\left(\dfrac{769}{18}\right)\sqrt{5}-\dfrac{29003}{420}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{5}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{58}{5}\right)\sqrt{20}\right)-\dfrac{4}{3}+\left(\dfrac{61}{4}\right)\sqrt{25}+0+\left(-\dfrac{41}{8}\right)\sqrt{20}+\left(\dfrac{34}{9}\right)\sqrt{20}+\dfrac{19}{7}+0+\left(-\dfrac{15}{4}\right)\sqrt{25}+\left(-7\right)\sqrt{25}\right)\times\left(\left(-\dfrac{38}{5}\right)\sqrt{45}+\left(-\dfrac{23}{3}\right)\sqrt{125}+\left(-7\right)\sqrt{125}+\dfrac{62}{5}-8+\dfrac{29}{8}+\left(\dfrac{19}{3}\right)\sqrt{20}+\left(\dfrac{66}{7}\right)\sqrt{25}+\left(\dfrac{47}{4}\right)\sqrt{125}+\left(\left(\dfrac{41}{8}\right)\sqrt{25}\right)-\left(\left(-\dfrac{17}{7}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{116}{5}\right)\sqrt{5}\right)-\dfrac{4}{3}+\dfrac{305}{4}+0+\left(-\dfrac{41}{4}\right)\sqrt{5}+\left(\dfrac{68}{9}\right)\sqrt{5}+\dfrac{19}{7}+0-\dfrac{75}{4}-35\right)\times\left(\left(-\dfrac{114}{5}\right)\sqrt{5}+\left(-\dfrac{115}{3}\right)\sqrt{5}+\left(-35\right)\sqrt{5}+\dfrac{62}{5}-8+\dfrac{29}{8}+\left(\dfrac{38}{3}\right)\sqrt{5}+\dfrac{330}{7}+\left(\dfrac{235}{4}\right)\sqrt{5}+\dfrac{205}{8}+\dfrac{85}{7}\right)\\
&=&\left(\left(\dfrac{3241}{180}\right)\sqrt{5}+\dfrac{1003}{42}\right)\left(\left(-\dfrac{1483}{60}\right)\sqrt{5}+\dfrac{13011}{140}\right)\\
&=&\left(-\dfrac{4806403}{10800}\right)\sqrt{25}+\left(\dfrac{3275299320}{3024000}\right)\sqrt{5}+\dfrac{4350011}{1960}\\
&=&-\dfrac{613153}{105840}+\left(\dfrac{3275299320}{3024000}\right)\sqrt{5}\\
\end{eqnarray*}