L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(9\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{20}\right)+\left(\dfrac{67}{7}\right)\sqrt{20}+\left(0\right)\sqrt{20}+\left(-\dfrac{5}{2}\right)\sqrt{25}+\dfrac{20}{7}+\left(-\dfrac{23}{7}\right)\sqrt{25}\) et \( Y=\left(\left(-\dfrac{28}{9}\right)\sqrt{45}\right)-\left(\left(0\right)\sqrt{25}\right)-\left(\left(5\right)\sqrt{25}\right)+\dfrac{63}{2}-\left(\left(-\dfrac{71}{3}\right)\sqrt{45}\right)+\left(-\dfrac{69}{2}\right)\sqrt{25}+\left(\left(\dfrac{11}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{67}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{19}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{125}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(9\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{20}\right)+\left(\dfrac{67}{7}\right)\sqrt{20}+\left(0\right)\sqrt{20}+\left(-\dfrac{5}{2}\right)\sqrt{25}+\dfrac{20}{7}+\left(-\dfrac{23}{7}\right)\sqrt{25}\right)+\left(\left(\left(-\dfrac{28}{9}\right)\sqrt{45}\right)-\left(\left(0\right)\sqrt{25}\right)-\left(\left(5\right)\sqrt{25}\right)+\dfrac{63}{2}-\left(\left(-\dfrac{71}{3}\right)\sqrt{45}\right)+\left(-\dfrac{69}{2}\right)\sqrt{25}+\left(\left(\dfrac{11}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{67}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{19}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(18\right)\sqrt{5}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{5}\right)-\left(\left(5\right)\sqrt{5}\right)+\left(\dfrac{134}{7}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{25}{2}+\dfrac{20}{7}-\dfrac{115}{7}\right)+\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{5}\right)-0-25+\dfrac{63}{2}-\left(\left(-71\right)\sqrt{5}\right)-\dfrac{345}{2}+11-\left(\left(\dfrac{67}{2}\right)\sqrt{5}\right)+\dfrac{95}{9}-\left(\left(-13\right)\sqrt{5}\right)-\left(\left(\dfrac{200}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(18\right)\sqrt{5}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{5}\right)-\left(\left(5\right)\sqrt{5}\right)+\left(\dfrac{134}{7}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{25}{2}+\dfrac{20}{7}-\dfrac{115}{7}+\left(\left(-\dfrac{28}{3}\right)\sqrt{5}\right)-0-25+\dfrac{63}{2}-\left(\left(-71\right)\sqrt{5}\right)-\dfrac{345}{2}+11-\left(\left(\dfrac{67}{2}\right)\sqrt{5}\right)+\dfrac{95}{9}-\left(\left(-13\right)\sqrt{5}\right)-\left(\left(\dfrac{200}{3}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{225}{7}\right)\sqrt{5}-\dfrac{21485}{126}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(9\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{20}\right)+\left(\dfrac{67}{7}\right)\sqrt{20}+\left(0\right)\sqrt{20}+\left(-\dfrac{5}{2}\right)\sqrt{25}+\dfrac{20}{7}+\left(-\dfrac{23}{7}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{28}{9}\right)\sqrt{45}\right)-\left(\left(0\right)\sqrt{25}\right)-\left(\left(5\right)\sqrt{25}\right)+\dfrac{63}{2}-\left(\left(-\dfrac{71}{3}\right)\sqrt{45}\right)+\left(-\dfrac{69}{2}\right)\sqrt{25}+\left(\left(\dfrac{11}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{67}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{19}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(18\right)\sqrt{5}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{5}\right)-\left(\left(5\right)\sqrt{5}\right)+\left(\dfrac{134}{7}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{25}{2}+\dfrac{20}{7}-\dfrac{115}{7}\right)-\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{5}\right)-0-25+\dfrac{63}{2}-\left(\left(-71\right)\sqrt{5}\right)-\dfrac{345}{2}+11-\left(\left(\dfrac{67}{2}\right)\sqrt{5}\right)+\dfrac{95}{9}-\left(\left(-13\right)\sqrt{5}\right)-\left(\left(\dfrac{200}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{807}{14}\right)\sqrt{5}-\dfrac{365}{14}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{5}-\dfrac{1300}{9}\right)\\
&=&\left(\dfrac{807}{14}\right)\sqrt{5}-\dfrac{365}{14}+\left(\dfrac{51}{2}\right)\sqrt{5}+\dfrac{1300}{9}\\
&=&\left(\dfrac{582}{7}\right)\sqrt{5}+\dfrac{14915}{126}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(9\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{20}\right)+\left(\dfrac{67}{7}\right)\sqrt{20}+\left(0\right)\sqrt{20}+\left(-\dfrac{5}{2}\right)\sqrt{25}+\dfrac{20}{7}+\left(-\dfrac{23}{7}\right)\sqrt{25}\right)\times\left(\left(\left(-\dfrac{28}{9}\right)\sqrt{45}\right)-\left(\left(0\right)\sqrt{25}\right)-\left(\left(5\right)\sqrt{25}\right)+\dfrac{63}{2}-\left(\left(-\dfrac{71}{3}\right)\sqrt{45}\right)+\left(-\dfrac{69}{2}\right)\sqrt{25}+\left(\left(\dfrac{11}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{67}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{19}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(18\right)\sqrt{5}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{5}\right)-\left(\left(5\right)\sqrt{5}\right)+\left(\dfrac{134}{7}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{25}{2}+\dfrac{20}{7}-\dfrac{115}{7}\right)\times\left(\left(\left(-\dfrac{28}{3}\right)\sqrt{5}\right)-0-25+\dfrac{63}{2}-\left(\left(-71\right)\sqrt{5}\right)-\dfrac{345}{2}+11-\left(\left(\dfrac{67}{2}\right)\sqrt{5}\right)+\dfrac{95}{9}-\left(\left(-13\right)\sqrt{5}\right)-\left(\left(\dfrac{200}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{807}{14}\right)\sqrt{5}-\dfrac{365}{14}\right)\left(\left(-\dfrac{51}{2}\right)\sqrt{5}-\dfrac{1300}{9}\right)\\
&=&\left(-\dfrac{41157}{28}\right)\sqrt{25}+\left(-\dfrac{643555}{84}\right)\sqrt{5}+\dfrac{237250}{63}\\
&=&-\dfrac{903065}{252}+\left(-\dfrac{643555}{84}\right)\sqrt{5}\\
\end{eqnarray*}