L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-1\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{37}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{47}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{20}\right)-\left(\left(1\right)\sqrt{20}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{25}\right)\right)\) et \( Y=\left(\dfrac{26}{7}-\dfrac{15}{8}+\left(\dfrac{73}{5}\right)\sqrt{125}-\dfrac{38}{9}-\dfrac{19}{3}\right)-\left(\left(-7\right)\sqrt{25}+\left(-\dfrac{29}{6}\right)\sqrt{25}+\left(-7\right)\sqrt{25}+\left(\dfrac{26}{5}\right)\sqrt{45}+\left(-\dfrac{15}{2}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{45}\right)-\left(\left(-\dfrac{67}{9}\right)\sqrt{45}\right)-\dfrac{11}{4}\) . 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(-1\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{37}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{47}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{20}\right)-\left(\left(1\right)\sqrt{20}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{25}\right)\right)\right)+\left(\left(\dfrac{26}{7}-\dfrac{15}{8}+\left(\dfrac{73}{5}\right)\sqrt{125}-\dfrac{38}{9}-\dfrac{19}{3}\right)-\left(\left(-7\right)\sqrt{25}+\left(-\dfrac{29}{6}\right)\sqrt{25}+\left(-7\right)\sqrt{25}+\left(\dfrac{26}{5}\right)\sqrt{45}+\left(-\dfrac{15}{2}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{45}\right)-\left(\left(-\dfrac{67}{9}\right)\sqrt{45}\right)-\dfrac{11}{4}\right)\\
&=&\left(\left(\left(-3\right)\sqrt{5}\right)-\left(-37-\left(\left(-\dfrac{235}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{5}\right)-\left(\left(2\right)\sqrt{5}\right)-\dfrac{315}{8}\right)\right)+\left(\left(\dfrac{26}{7}-\dfrac{15}{8}+\left(73\right)\sqrt{5}-\dfrac{38}{9}-\dfrac{19}{3}\right)-\left(-35-\dfrac{145}{6}-35+\left(\dfrac{78}{5}\right)\sqrt{5}+\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-\left(\left(6\right)\sqrt{5}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{5}\right)-\dfrac{11}{4}\right)\\
&=&\left(\left(-3\right)\sqrt{5}\right)-\left(-37-\left(\left(-\dfrac{235}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{5}\right)-\left(\left(2\right)\sqrt{5}\right)-\dfrac{315}{8}\right)+\left(\dfrac{26}{7}-\dfrac{15}{8}+\left(73\right)\sqrt{5}-\dfrac{38}{9}-\dfrac{19}{3}\right)-\left(-35-\dfrac{145}{6}-35+\left(\dfrac{78}{5}\right)\sqrt{5}+\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-\left(\left(6\right)\sqrt{5}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{5}\right)-\dfrac{11}{4}\\
&=&\left(\dfrac{2159}{60}\right)\sqrt{5}+\dfrac{40087}{252}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-1\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{37}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{47}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{20}\right)-\left(\left(1\right)\sqrt{20}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{25}\right)\right)\right)-\left(\left(\dfrac{26}{7}-\dfrac{15}{8}+\left(\dfrac{73}{5}\right)\sqrt{125}-\dfrac{38}{9}-\dfrac{19}{3}\right)-\left(\left(-7\right)\sqrt{25}+\left(-\dfrac{29}{6}\right)\sqrt{25}+\left(-7\right)\sqrt{25}+\left(\dfrac{26}{5}\right)\sqrt{45}+\left(-\dfrac{15}{2}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{45}\right)-\left(\left(-\dfrac{67}{9}\right)\sqrt{45}\right)-\dfrac{11}{4}\right)\\
&=&\left(\left(\left(-3\right)\sqrt{5}\right)-\left(-37-\left(\left(-\dfrac{235}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{5}\right)-\left(\left(2\right)\sqrt{5}\right)-\dfrac{315}{8}\right)\right)-\left(\left(\dfrac{26}{7}-\dfrac{15}{8}+\left(73\right)\sqrt{5}-\dfrac{38}{9}-\dfrac{19}{3}\right)-\left(-35-\dfrac{145}{6}-35+\left(\dfrac{78}{5}\right)\sqrt{5}+\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-\left(\left(6\right)\sqrt{5}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{5}\right)-\dfrac{11}{4}\right)\\
&=&\left(\left(-\dfrac{241}{4}\right)\sqrt{5}+\dfrac{611}{8}\right)-\left(\dfrac{41681}{504}+\left(\dfrac{2887}{30}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{241}{4}\right)\sqrt{5}+\dfrac{611}{8}+-\dfrac{41681}{504}+\left(-\dfrac{2887}{30}\right)\sqrt{5}\\
&=&\left(-\dfrac{9389}{60}\right)\sqrt{5}-\dfrac{797}{126}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-1\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{37}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{47}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{20}\right)-\left(\left(1\right)\sqrt{20}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{25}\right)\right)\right)\times\left(\left(\dfrac{26}{7}-\dfrac{15}{8}+\left(\dfrac{73}{5}\right)\sqrt{125}-\dfrac{38}{9}-\dfrac{19}{3}\right)-\left(\left(-7\right)\sqrt{25}+\left(-\dfrac{29}{6}\right)\sqrt{25}+\left(-7\right)\sqrt{25}+\left(\dfrac{26}{5}\right)\sqrt{45}+\left(-\dfrac{15}{2}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{45}\right)-\left(\left(-\dfrac{67}{9}\right)\sqrt{45}\right)-\dfrac{11}{4}\right)\\
&=&\left(\left(\left(-3\right)\sqrt{5}\right)-\left(-37-\left(\left(-\dfrac{235}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{5}\right)-\left(\left(2\right)\sqrt{5}\right)-\dfrac{315}{8}\right)\right)\times\left(\left(\dfrac{26}{7}-\dfrac{15}{8}+\left(73\right)\sqrt{5}-\dfrac{38}{9}-\dfrac{19}{3}\right)-\left(-35-\dfrac{145}{6}-35+\left(\dfrac{78}{5}\right)\sqrt{5}+\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-\left(\left(6\right)\sqrt{5}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{5}\right)-\dfrac{11}{4}\right)\\
&=&\left(\left(-\dfrac{241}{4}\right)\sqrt{5}+\dfrac{611}{8}\right)\left(\dfrac{41681}{504}+\left(\dfrac{2887}{30}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{23860589}{10080}\right)\sqrt{5}+\left(-\dfrac{695767}{120}\right)\sqrt{25}+\dfrac{25467091}{4032}\\
&=&\left(\dfrac{23860589}{10080}\right)\sqrt{5}-\dfrac{2194122360}{96768}\\
\end{eqnarray*}