L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{74}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{20}\right)-1\right)+\dfrac{77}{6}-\left(\left(-5\right)\sqrt{20}+\dfrac{69}{2}+\dfrac{33}{2}\right)\) et \( Y=\left(\left(9\right)\sqrt{125}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\left(\dfrac{5}{4}+\dfrac{3}{2}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{20}\right)\) . 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(\left(\dfrac{74}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{20}\right)-1\right)+\dfrac{77}{6}-\left(\left(-5\right)\sqrt{20}+\dfrac{69}{2}+\dfrac{33}{2}\right)\right)+\left(\left(\left(9\right)\sqrt{125}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\left(\dfrac{5}{4}+\dfrac{3}{2}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{148}{7}\right)\sqrt{5}\right)-\left(\left(39\right)\sqrt{5}\right)-1\right)+\dfrac{77}{6}-\left(\left(-10\right)\sqrt{5}+\dfrac{69}{2}+\dfrac{33}{2}\right)\right)+\left(\left(\left(45\right)\sqrt{5}\right)-\left(\left(27\right)\sqrt{5}\right)-\left(\dfrac{5}{4}+\dfrac{3}{2}\right)-\left(\left(-21\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\left(\dfrac{148}{7}\right)\sqrt{5}\right)-\left(\left(39\right)\sqrt{5}\right)-1\right)+\dfrac{77}{6}-\left(\left(-10\right)\sqrt{5}+\dfrac{69}{2}+\dfrac{33}{2}\right)+\left(\left(45\right)\sqrt{5}\right)-\left(\left(27\right)\sqrt{5}\right)-\left(\dfrac{5}{4}+\dfrac{3}{2}\right)-\left(\left(-21\right)\sqrt{5}\right)\\
&=&\left(\dfrac{218}{7}\right)\sqrt{5}-\dfrac{503}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{74}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{20}\right)-1\right)+\dfrac{77}{6}-\left(\left(-5\right)\sqrt{20}+\dfrac{69}{2}+\dfrac{33}{2}\right)\right)-\left(\left(\left(9\right)\sqrt{125}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\left(\dfrac{5}{4}+\dfrac{3}{2}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{148}{7}\right)\sqrt{5}\right)-\left(\left(39\right)\sqrt{5}\right)-1\right)+\dfrac{77}{6}-\left(\left(-10\right)\sqrt{5}+\dfrac{69}{2}+\dfrac{33}{2}\right)\right)-\left(\left(\left(45\right)\sqrt{5}\right)-\left(\left(27\right)\sqrt{5}\right)-\left(\dfrac{5}{4}+\dfrac{3}{2}\right)-\left(\left(-21\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{55}{7}\right)\sqrt{5}-\dfrac{235}{6}\right)-\left(\left(39\right)\sqrt{5}-\dfrac{11}{4}\right)\\
&=&\left(-\dfrac{55}{7}\right)\sqrt{5}-\dfrac{235}{6}+\left(-39\right)\sqrt{5}+\dfrac{11}{4}\\
&=&\left(-\dfrac{328}{7}\right)\sqrt{5}-\dfrac{437}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{74}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{20}\right)-1\right)+\dfrac{77}{6}-\left(\left(-5\right)\sqrt{20}+\dfrac{69}{2}+\dfrac{33}{2}\right)\right)\times\left(\left(\left(9\right)\sqrt{125}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\left(\dfrac{5}{4}+\dfrac{3}{2}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{148}{7}\right)\sqrt{5}\right)-\left(\left(39\right)\sqrt{5}\right)-1\right)+\dfrac{77}{6}-\left(\left(-10\right)\sqrt{5}+\dfrac{69}{2}+\dfrac{33}{2}\right)\right)\times\left(\left(\left(45\right)\sqrt{5}\right)-\left(\left(27\right)\sqrt{5}\right)-\left(\dfrac{5}{4}+\dfrac{3}{2}\right)-\left(\left(-21\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{55}{7}\right)\sqrt{5}-\dfrac{235}{6}\right)\left(\left(39\right)\sqrt{5}-\dfrac{11}{4}\right)\\
&=&\left(-\dfrac{2145}{7}\right)\sqrt{25}+\left(-\dfrac{42165}{28}\right)\sqrt{5}+\dfrac{2585}{24}\\
&=&-\dfrac{239305}{168}+\left(-\dfrac{42165}{28}\right)\sqrt{5}\\
\end{eqnarray*}