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Exercice
Soit \( X=\left(\dfrac{23}{4}\right)\sqrt{45}+\left(-\dfrac{19}{5}\right)\sqrt{125}-\dfrac{19}{5}+\left(-\dfrac{1}{2}\right)\sqrt{125}+\left(-\dfrac{5}{3}\right)\sqrt{125}-4+\left(-\dfrac{44}{7}\right)\sqrt{125}+\left(-\dfrac{69}{5}\right)\sqrt{20}+\left(-9\right)\sqrt{20}\) et \( Y=\dfrac{71}{5}+\left(-\dfrac{23}{2}\right)\sqrt{125}+\left(-7\right)\sqrt{20}+\left(\dfrac{61}{8}\right)\sqrt{20}-\dfrac{22}{3}+2+\left(\dfrac{14}{5}\right)\sqrt{20}+\left(\dfrac{13}{2}\right)\sqrt{25}-\dfrac{35}{2}+\left(\dfrac{33}{8}\right)\sqrt{45}+\left(-\dfrac{23}{2}\right)\sqrt{125}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{23}{4}\right)\sqrt{45}+\left(-\dfrac{19}{5}\right)\sqrt{125}-\dfrac{19}{5}+\left(-\dfrac{1}{2}\right)\sqrt{125}+\left(-\dfrac{5}{3}\right)\sqrt{125}-4+\left(-\dfrac{44}{7}\right)\sqrt{125}+\left(-\dfrac{69}{5}\right)\sqrt{20}+\left(-9\right)\sqrt{20}\right)+\left(\dfrac{71}{5}+\left(-\dfrac{23}{2}\right)\sqrt{125}+\left(-7\right)\sqrt{20}+\left(\dfrac{61}{8}\right)\sqrt{20}-\dfrac{22}{3}+2+\left(\dfrac{14}{5}\right)\sqrt{20}+\left(\dfrac{13}{2}\right)\sqrt{25}-\dfrac{35}{2}+\left(\dfrac{33}{8}\right)\sqrt{45}+\left(-\dfrac{23}{2}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{69}{4}\right)\sqrt{5}+\left(-19\right)\sqrt{5}-\dfrac{19}{5}+\left(-\dfrac{5}{2}\right)\sqrt{5}+\left(-\dfrac{25}{3}\right)\sqrt{5}-4+\left(-\dfrac{220}{7}\right)\sqrt{5}+\left(-\dfrac{138}{5}\right)\sqrt{5}+\left(-18\right)\sqrt{5}\right)+\left(\dfrac{71}{5}+\left(-\dfrac{115}{2}\right)\sqrt{5}+\left(-14\right)\sqrt{5}+\left(\dfrac{61}{4}\right)\sqrt{5}-\dfrac{22}{3}+2+\left(\dfrac{28}{5}\right)\sqrt{5}+\dfrac{65}{2}-\dfrac{35}{2}+\left(\dfrac{99}{8}\right)\sqrt{5}+\left(-\dfrac{115}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{69}{4}\right)\sqrt{5}+\left(-19\right)\sqrt{5}-\dfrac{19}{5}+\left(-\dfrac{5}{2}\right)\sqrt{5}+\left(-\dfrac{25}{3}\right)\sqrt{5}-4+\left(-\dfrac{220}{7}\right)\sqrt{5}+\left(-\dfrac{138}{5}\right)\sqrt{5}+\left(-18\right)\sqrt{5}+\dfrac{71}{5}+\left(-\dfrac{115}{2}\right)\sqrt{5}+\left(-14\right)\sqrt{5}+\left(\dfrac{61}{4}\right)\sqrt{5}-\dfrac{22}{3}+2+\left(\dfrac{28}{5}\right)\sqrt{5}+\dfrac{65}{2}-\dfrac{35}{2}+\left(\dfrac{99}{8}\right)\sqrt{5}+\left(-\dfrac{115}{2}\right)\sqrt{5}\\
&=&\left(-\dfrac{31145}{168}\right)\sqrt{5}+\dfrac{241}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{23}{4}\right)\sqrt{45}+\left(-\dfrac{19}{5}\right)\sqrt{125}-\dfrac{19}{5}+\left(-\dfrac{1}{2}\right)\sqrt{125}+\left(-\dfrac{5}{3}\right)\sqrt{125}-4+\left(-\dfrac{44}{7}\right)\sqrt{125}+\left(-\dfrac{69}{5}\right)\sqrt{20}+\left(-9\right)\sqrt{20}\right)-\left(\dfrac{71}{5}+\left(-\dfrac{23}{2}\right)\sqrt{125}+\left(-7\right)\sqrt{20}+\left(\dfrac{61}{8}\right)\sqrt{20}-\dfrac{22}{3}+2+\left(\dfrac{14}{5}\right)\sqrt{20}+\left(\dfrac{13}{2}\right)\sqrt{25}-\dfrac{35}{2}+\left(\dfrac{33}{8}\right)\sqrt{45}+\left(-\dfrac{23}{2}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{69}{4}\right)\sqrt{5}+\left(-19\right)\sqrt{5}-\dfrac{19}{5}+\left(-\dfrac{5}{2}\right)\sqrt{5}+\left(-\dfrac{25}{3}\right)\sqrt{5}-4+\left(-\dfrac{220}{7}\right)\sqrt{5}+\left(-\dfrac{138}{5}\right)\sqrt{5}+\left(-18\right)\sqrt{5}\right)-\left(\dfrac{71}{5}+\left(-\dfrac{115}{2}\right)\sqrt{5}+\left(-14\right)\sqrt{5}+\left(\dfrac{61}{4}\right)\sqrt{5}-\dfrac{22}{3}+2+\left(\dfrac{28}{5}\right)\sqrt{5}+\dfrac{65}{2}-\dfrac{35}{2}+\left(\dfrac{99}{8}\right)\sqrt{5}+\left(-\dfrac{115}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{37637}{420}\right)\sqrt{5}-\dfrac{39}{5}\right)-\left(\dfrac{358}{15}+\left(-\dfrac{3831}{40}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{37637}{420}\right)\sqrt{5}-\dfrac{39}{5}+-\dfrac{358}{15}+\left(\dfrac{3831}{40}\right)\sqrt{5}\\
&=&\left(\dfrac{5177}{840}\right)\sqrt{5}-\dfrac{95}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{23}{4}\right)\sqrt{45}+\left(-\dfrac{19}{5}\right)\sqrt{125}-\dfrac{19}{5}+\left(-\dfrac{1}{2}\right)\sqrt{125}+\left(-\dfrac{5}{3}\right)\sqrt{125}-4+\left(-\dfrac{44}{7}\right)\sqrt{125}+\left(-\dfrac{69}{5}\right)\sqrt{20}+\left(-9\right)\sqrt{20}\right)\times\left(\dfrac{71}{5}+\left(-\dfrac{23}{2}\right)\sqrt{125}+\left(-7\right)\sqrt{20}+\left(\dfrac{61}{8}\right)\sqrt{20}-\dfrac{22}{3}+2+\left(\dfrac{14}{5}\right)\sqrt{20}+\left(\dfrac{13}{2}\right)\sqrt{25}-\dfrac{35}{2}+\left(\dfrac{33}{8}\right)\sqrt{45}+\left(-\dfrac{23}{2}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{69}{4}\right)\sqrt{5}+\left(-19\right)\sqrt{5}-\dfrac{19}{5}+\left(-\dfrac{5}{2}\right)\sqrt{5}+\left(-\dfrac{25}{3}\right)\sqrt{5}-4+\left(-\dfrac{220}{7}\right)\sqrt{5}+\left(-\dfrac{138}{5}\right)\sqrt{5}+\left(-18\right)\sqrt{5}\right)\times\left(\dfrac{71}{5}+\left(-\dfrac{115}{2}\right)\sqrt{5}+\left(-14\right)\sqrt{5}+\left(\dfrac{61}{4}\right)\sqrt{5}-\dfrac{22}{3}+2+\left(\dfrac{28}{5}\right)\sqrt{5}+\dfrac{65}{2}-\dfrac{35}{2}+\left(\dfrac{99}{8}\right)\sqrt{5}+\left(-\dfrac{115}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{37637}{420}\right)\sqrt{5}-\dfrac{39}{5}\right)\left(\dfrac{358}{15}+\left(-\dfrac{3831}{40}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{701413}{504}\right)\sqrt{5}+\left(\dfrac{48062449}{5600}\right)\sqrt{25}-\dfrac{4654}{25}\\
&=&\left(-\dfrac{701413}{504}\right)\sqrt{5}+\dfrac{239269749}{5600}\\
\end{eqnarray*}