L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-9-\left(\left(-\dfrac{49}{3}\right)\sqrt{20}\right)\right)-\left(\left(6\right)\sqrt{25}+\left(\dfrac{29}{2}\right)\sqrt{45}\right)+\dfrac{39}{5}\) et \( Y=\dfrac{46}{5}-\left(\left(2\right)\sqrt{125}\right)-\dfrac{81}{7}+\dfrac{29}{8}+\left(-\dfrac{77}{3}\right)\sqrt{45}+\left(-\dfrac{35}{2}\right)\sqrt{125}-5\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-9-\left(\left(-\dfrac{49}{3}\right)\sqrt{20}\right)\right)-\left(\left(6\right)\sqrt{25}+\left(\dfrac{29}{2}\right)\sqrt{45}\right)+\dfrac{39}{5}\right)+\left(\dfrac{46}{5}-\left(\left(2\right)\sqrt{125}\right)-\dfrac{81}{7}+\dfrac{29}{8}+\left(-\dfrac{77}{3}\right)\sqrt{45}+\left(-\dfrac{35}{2}\right)\sqrt{125}-5\right)\\
&=&\left(\left(-9-\left(\left(-\dfrac{98}{3}\right)\sqrt{5}\right)\right)-\left(30+\left(\dfrac{87}{2}\right)\sqrt{5}\right)+\dfrac{39}{5}\right)+\left(\dfrac{46}{5}-\left(\left(10\right)\sqrt{5}\right)-\dfrac{81}{7}+\dfrac{29}{8}+\left(-77\right)\sqrt{5}+\left(-\dfrac{175}{2}\right)\sqrt{5}-5\right)\\
&=&\left(-9-\left(\left(-\dfrac{98}{3}\right)\sqrt{5}\right)\right)-\left(30+\left(\dfrac{87}{2}\right)\sqrt{5}\right)+\dfrac{39}{5}+\dfrac{46}{5}-\left(\left(10\right)\sqrt{5}\right)-\dfrac{81}{7}+\dfrac{29}{8}+\left(-77\right)\sqrt{5}+\left(-\dfrac{175}{2}\right)\sqrt{5}-5\\
&=&-\dfrac{1957}{56}+\left(-\dfrac{556}{3}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-9-\left(\left(-\dfrac{49}{3}\right)\sqrt{20}\right)\right)-\left(\left(6\right)\sqrt{25}+\left(\dfrac{29}{2}\right)\sqrt{45}\right)+\dfrac{39}{5}\right)-\left(\dfrac{46}{5}-\left(\left(2\right)\sqrt{125}\right)-\dfrac{81}{7}+\dfrac{29}{8}+\left(-\dfrac{77}{3}\right)\sqrt{45}+\left(-\dfrac{35}{2}\right)\sqrt{125}-5\right)\\
&=&\left(\left(-9-\left(\left(-\dfrac{98}{3}\right)\sqrt{5}\right)\right)-\left(30+\left(\dfrac{87}{2}\right)\sqrt{5}\right)+\dfrac{39}{5}\right)-\left(\dfrac{46}{5}-\left(\left(10\right)\sqrt{5}\right)-\dfrac{81}{7}+\dfrac{29}{8}+\left(-77\right)\sqrt{5}+\left(-\dfrac{175}{2}\right)\sqrt{5}-5\right)\\
&=&\left(-\dfrac{156}{5}+\left(-\dfrac{65}{6}\right)\sqrt{5}\right)-\left(-\dfrac{1049}{280}+\left(-\dfrac{349}{2}\right)\sqrt{5}\right)\\
&=&-\dfrac{156}{5}+\left(-\dfrac{65}{6}\right)\sqrt{5}+\dfrac{1049}{280}+\left(\dfrac{349}{2}\right)\sqrt{5}\\
&=&-\dfrac{7687}{280}+\left(\dfrac{491}{3}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-9-\left(\left(-\dfrac{49}{3}\right)\sqrt{20}\right)\right)-\left(\left(6\right)\sqrt{25}+\left(\dfrac{29}{2}\right)\sqrt{45}\right)+\dfrac{39}{5}\right)\times\left(\dfrac{46}{5}-\left(\left(2\right)\sqrt{125}\right)-\dfrac{81}{7}+\dfrac{29}{8}+\left(-\dfrac{77}{3}\right)\sqrt{45}+\left(-\dfrac{35}{2}\right)\sqrt{125}-5\right)\\
&=&\left(\left(-9-\left(\left(-\dfrac{98}{3}\right)\sqrt{5}\right)\right)-\left(30+\left(\dfrac{87}{2}\right)\sqrt{5}\right)+\dfrac{39}{5}\right)\times\left(\dfrac{46}{5}-\left(\left(10\right)\sqrt{5}\right)-\dfrac{81}{7}+\dfrac{29}{8}+\left(-77\right)\sqrt{5}+\left(-\dfrac{175}{2}\right)\sqrt{5}-5\right)\\
&=&\left(-\dfrac{156}{5}+\left(-\dfrac{65}{6}\right)\sqrt{5}\right)\left(-\dfrac{1049}{280}+\left(-\dfrac{349}{2}\right)\sqrt{5}\right)\\
&=&\dfrac{40911}{350}+\left(\dfrac{9214777}{1680}\right)\sqrt{5}+\left(\dfrac{22685}{12}\right)\sqrt{25}\\
&=&\dfrac{20094841}{2100}+\left(\dfrac{9214777}{1680}\right)\sqrt{5}\\
\end{eqnarray*}