L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=-\dfrac{16}{7}+\left(\left(0\right)\sqrt{12}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{9}\right)-\left(\left(2\right)\sqrt{9}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{27}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{27}\right)+\left(\dfrac{25}{2}\right)\sqrt{9}+\left(\dfrac{16}{3}\right)\sqrt{12}+\left(-\dfrac{70}{3}\right)\sqrt{12}+\left(-\dfrac{51}{5}\right)\sqrt{9}+\left(-\dfrac{66}{5}\right)\sqrt{12}+\left(4\right)\sqrt{75}+\left(\dfrac{62}{7}\right)\sqrt{27}-\dfrac{62}{9}-\left(\left(-\dfrac{4}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{64}{9}\right)\sqrt{9}\right)\) et \( Y=\left(\left(-8\right)\sqrt{75}+\dfrac{44}{5}+\left(-\dfrac{76}{7}\right)\sqrt{9}+\left(\dfrac{66}{5}\right)\sqrt{12}\right)-9-\dfrac{80}{7}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{16}{7}+\left(\left(0\right)\sqrt{12}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{9}\right)-\left(\left(2\right)\sqrt{9}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{27}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{27}\right)+\left(\dfrac{25}{2}\right)\sqrt{9}+\left(\dfrac{16}{3}\right)\sqrt{12}+\left(-\dfrac{70}{3}\right)\sqrt{12}+\left(-\dfrac{51}{5}\right)\sqrt{9}+\left(-\dfrac{66}{5}\right)\sqrt{12}+\left(4\right)\sqrt{75}+\left(\dfrac{62}{7}\right)\sqrt{27}-\dfrac{62}{9}-\left(\left(-\dfrac{4}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{64}{9}\right)\sqrt{9}\right)\right)+\left(\left(\left(-8\right)\sqrt{75}+\dfrac{44}{5}+\left(-\dfrac{76}{7}\right)\sqrt{9}+\left(\dfrac{66}{5}\right)\sqrt{12}\right)-9-\dfrac{80}{7}\right)\\
&=&\left(-\dfrac{16}{7}+\left(\left(0\right)\sqrt{3}\right)-\dfrac{207}{2}-6-\left(\left(\dfrac{195}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{195}{8}\right)\sqrt{3}\right)+\dfrac{75}{2}+\left(\dfrac{32}{3}\right)\sqrt{3}+\left(-\dfrac{140}{3}\right)\sqrt{3}-\dfrac{153}{5}+\left(-\dfrac{132}{5}\right)\sqrt{3}+\left(20\right)\sqrt{3}+\left(\dfrac{186}{7}\right)\sqrt{3}-\dfrac{62}{9}+\dfrac{12}{5}-\dfrac{64}{3}\right)+\left(\left(\left(-40\right)\sqrt{3}+\dfrac{44}{5}-\dfrac{228}{7}+\left(\dfrac{132}{5}\right)\sqrt{3}\right)-9-\dfrac{80}{7}\right)\\
&=&-\dfrac{16}{7}+\left(\left(0\right)\sqrt{3}\right)-\dfrac{207}{2}-6-\left(\left(\dfrac{195}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{195}{8}\right)\sqrt{3}\right)+\dfrac{75}{2}+\left(\dfrac{32}{3}\right)\sqrt{3}+\left(-\dfrac{140}{3}\right)\sqrt{3}-\dfrac{153}{5}+\left(-\dfrac{132}{5}\right)\sqrt{3}+\left(20\right)\sqrt{3}+\left(\dfrac{186}{7}\right)\sqrt{3}-\dfrac{62}{9}+\dfrac{12}{5}-\dfrac{64}{3}+\left(\left(-40\right)\sqrt{3}+\dfrac{44}{5}-\dfrac{228}{7}+\left(\dfrac{132}{5}\right)\sqrt{3}\right)-9-\dfrac{80}{7}\\
&=&-\dfrac{55096}{315}+\left(-\dfrac{2189}{28}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{16}{7}+\left(\left(0\right)\sqrt{12}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{9}\right)-\left(\left(2\right)\sqrt{9}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{27}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{27}\right)+\left(\dfrac{25}{2}\right)\sqrt{9}+\left(\dfrac{16}{3}\right)\sqrt{12}+\left(-\dfrac{70}{3}\right)\sqrt{12}+\left(-\dfrac{51}{5}\right)\sqrt{9}+\left(-\dfrac{66}{5}\right)\sqrt{12}+\left(4\right)\sqrt{75}+\left(\dfrac{62}{7}\right)\sqrt{27}-\dfrac{62}{9}-\left(\left(-\dfrac{4}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{64}{9}\right)\sqrt{9}\right)\right)-\left(\left(\left(-8\right)\sqrt{75}+\dfrac{44}{5}+\left(-\dfrac{76}{7}\right)\sqrt{9}+\left(\dfrac{66}{5}\right)\sqrt{12}\right)-9-\dfrac{80}{7}\right)\\
&=&\left(-\dfrac{16}{7}+\left(\left(0\right)\sqrt{3}\right)-\dfrac{207}{2}-6-\left(\left(\dfrac{195}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{195}{8}\right)\sqrt{3}\right)+\dfrac{75}{2}+\left(\dfrac{32}{3}\right)\sqrt{3}+\left(-\dfrac{140}{3}\right)\sqrt{3}-\dfrac{153}{5}+\left(-\dfrac{132}{5}\right)\sqrt{3}+\left(20\right)\sqrt{3}+\left(\dfrac{186}{7}\right)\sqrt{3}-\dfrac{62}{9}+\dfrac{12}{5}-\dfrac{64}{3}\right)-\left(\left(\left(-40\right)\sqrt{3}+\dfrac{44}{5}-\dfrac{228}{7}+\left(\dfrac{132}{5}\right)\sqrt{3}\right)-9-\dfrac{80}{7}\right)\\
&=&\left(-\dfrac{41173}{315}+\left(-\dfrac{9041}{140}\right)\sqrt{3}\right)-\left(\left(-\dfrac{68}{5}\right)\sqrt{3}-\dfrac{221}{5}\right)\\
&=&-\dfrac{41173}{315}+\left(-\dfrac{9041}{140}\right)\sqrt{3}+\left(\dfrac{68}{5}\right)\sqrt{3}+\dfrac{221}{5}\\
&=&-\dfrac{5450}{63}+\left(-\dfrac{7137}{140}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{16}{7}+\left(\left(0\right)\sqrt{12}\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{9}\right)-\left(\left(2\right)\sqrt{9}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{27}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{27}\right)+\left(\dfrac{25}{2}\right)\sqrt{9}+\left(\dfrac{16}{3}\right)\sqrt{12}+\left(-\dfrac{70}{3}\right)\sqrt{12}+\left(-\dfrac{51}{5}\right)\sqrt{9}+\left(-\dfrac{66}{5}\right)\sqrt{12}+\left(4\right)\sqrt{75}+\left(\dfrac{62}{7}\right)\sqrt{27}-\dfrac{62}{9}-\left(\left(-\dfrac{4}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{64}{9}\right)\sqrt{9}\right)\right)\times\left(\left(\left(-8\right)\sqrt{75}+\dfrac{44}{5}+\left(-\dfrac{76}{7}\right)\sqrt{9}+\left(\dfrac{66}{5}\right)\sqrt{12}\right)-9-\dfrac{80}{7}\right)\\
&=&\left(-\dfrac{16}{7}+\left(\left(0\right)\sqrt{3}\right)-\dfrac{207}{2}-6-\left(\left(\dfrac{195}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{195}{8}\right)\sqrt{3}\right)+\dfrac{75}{2}+\left(\dfrac{32}{3}\right)\sqrt{3}+\left(-\dfrac{140}{3}\right)\sqrt{3}-\dfrac{153}{5}+\left(-\dfrac{132}{5}\right)\sqrt{3}+\left(20\right)\sqrt{3}+\left(\dfrac{186}{7}\right)\sqrt{3}-\dfrac{62}{9}+\dfrac{12}{5}-\dfrac{64}{3}\right)\times\left(\left(\left(-40\right)\sqrt{3}+\dfrac{44}{5}-\dfrac{228}{7}+\left(\dfrac{132}{5}\right)\sqrt{3}\right)-9-\dfrac{80}{7}\right)\\
&=&\left(-\dfrac{41173}{315}+\left(-\dfrac{9041}{140}\right)\sqrt{3}\right)\left(\left(-\dfrac{68}{5}\right)\sqrt{3}-\dfrac{221}{5}\right)\\
&=&\left(\dfrac{5106780875}{1102500}\right)\sqrt{3}+\dfrac{9099233}{1575}+\left(\dfrac{153697}{175}\right)\sqrt{9}\\
&=&\left(\dfrac{5106780875}{1102500}\right)\sqrt{3}+\dfrac{2318584100}{275625}\\
\end{eqnarray*}