L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{45}\right)-\left(\left(-1\right)\sqrt{45}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(3\right)\sqrt{20}+\dfrac{67}{8}+\left(7\right)\sqrt{20}+\left(-\dfrac{4}{3}\right)\sqrt{25}+\left(\dfrac{4}{3}\right)\sqrt{20}\right)\) et \( Y=\left(-\dfrac{60}{7}\right)\sqrt{25}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{45}\right)-\left(\left(-1\right)\sqrt{45}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(3\right)\sqrt{20}+\dfrac{67}{8}+\left(7\right)\sqrt{20}+\left(-\dfrac{4}{3}\right)\sqrt{25}+\left(\dfrac{4}{3}\right)\sqrt{20}\right)\right)+\left(\left(-\dfrac{60}{7}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(-3\right)\sqrt{5}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{5}\right)-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(6\right)\sqrt{5}+\dfrac{67}{8}+\left(14\right)\sqrt{5}-\dfrac{20}{3}+\left(\dfrac{8}{3}\right)\sqrt{5}\right)\right)+\left(-\dfrac{300}{7}\right)\\
&=&\left(\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(-3\right)\sqrt{5}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{5}\right)-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(6\right)\sqrt{5}+\dfrac{67}{8}+\left(14\right)\sqrt{5}-\dfrac{20}{3}+\left(\dfrac{8}{3}\right)\sqrt{5}\right)-\dfrac{300}{7}\\
&=&\left(-\dfrac{187}{2}\right)\sqrt{5}-\dfrac{7487}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{45}\right)-\left(\left(-1\right)\sqrt{45}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(3\right)\sqrt{20}+\dfrac{67}{8}+\left(7\right)\sqrt{20}+\left(-\dfrac{4}{3}\right)\sqrt{25}+\left(\dfrac{4}{3}\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{60}{7}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(-3\right)\sqrt{5}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{5}\right)-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(6\right)\sqrt{5}+\dfrac{67}{8}+\left(14\right)\sqrt{5}-\dfrac{20}{3}+\left(\dfrac{8}{3}\right)\sqrt{5}\right)\right)-\left(-\dfrac{300}{7}\right)\\
&=&\left(\left(-\dfrac{187}{2}\right)\sqrt{5}-\dfrac{41}{24}\right)-\left(-\dfrac{300}{7}\right)\\
&=&\left(-\dfrac{187}{2}\right)\sqrt{5}-\dfrac{41}{24}+\dfrac{300}{7}\\
&=&\left(-\dfrac{187}{2}\right)\sqrt{5}+\dfrac{6913}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{45}\right)-\left(\left(-1\right)\sqrt{45}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(3\right)\sqrt{20}+\dfrac{67}{8}+\left(7\right)\sqrt{20}+\left(-\dfrac{4}{3}\right)\sqrt{25}+\left(\dfrac{4}{3}\right)\sqrt{20}\right)\right)\times\left(\left(-\dfrac{60}{7}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\left(-\dfrac{33}{2}\right)\sqrt{5}\right)-\left(\left(-3\right)\sqrt{5}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{5}\right)-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(6\right)\sqrt{5}+\dfrac{67}{8}+\left(14\right)\sqrt{5}-\dfrac{20}{3}+\left(\dfrac{8}{3}\right)\sqrt{5}\right)\right)\times\left(-\dfrac{300}{7}\right)\\
&=&\left(\left(-\dfrac{187}{2}\right)\sqrt{5}-\dfrac{41}{24}\right)\left(-\dfrac{300}{7}\right)\\
&=&\left(\dfrac{28050}{7}\right)\sqrt{5}+\dfrac{1025}{14}\\
&=&\left(\dfrac{28050}{7}\right)\sqrt{5}+\dfrac{1025}{14}\\
\end{eqnarray*}