L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{11}{2}+\dfrac{33}{4}+\left(\dfrac{31}{2}\right)\sqrt{45}-3+\dfrac{27}{2}-\left(\left(\dfrac{81}{8}\right)\sqrt{25}\right)+\left(\dfrac{12}{5}\right)\sqrt{45}+\left(0\right)\sqrt{20}+\left(\dfrac{81}{8}\right)\sqrt{25}\) et \( Y=\left(\left(-\dfrac{52}{7}\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{29}{6}\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{33}{5}\right)\sqrt{20}\right)\right)-\left(\left(-7\right)\sqrt{45}+\left(-\dfrac{17}{2}\right)\sqrt{25}+\left(\dfrac{21}{5}\right)\sqrt{125}+\left(-\dfrac{71}{8}\right)\sqrt{25}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{11}{2}+\dfrac{33}{4}+\left(\dfrac{31}{2}\right)\sqrt{45}-3+\dfrac{27}{2}-\left(\left(\dfrac{81}{8}\right)\sqrt{25}\right)+\left(\dfrac{12}{5}\right)\sqrt{45}+\left(0\right)\sqrt{20}+\left(\dfrac{81}{8}\right)\sqrt{25}\right)+\left(\left(\left(-\dfrac{52}{7}\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{29}{6}\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{33}{5}\right)\sqrt{20}\right)\right)-\left(\left(-7\right)\sqrt{45}+\left(-\dfrac{17}{2}\right)\sqrt{25}+\left(\dfrac{21}{5}\right)\sqrt{125}+\left(-\dfrac{71}{8}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{11}{2}+\dfrac{33}{4}+\left(\dfrac{93}{2}\right)\sqrt{5}-3+\dfrac{27}{2}-\dfrac{405}{8}+\left(\dfrac{36}{5}\right)\sqrt{5}+\left(0\right)\sqrt{5}+\dfrac{405}{8}\right)+\left(\left(\left(-\dfrac{260}{7}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{29}{3}\right)\sqrt{5}\right)+\dfrac{85}{2}-\left(\left(-\dfrac{66}{5}\right)\sqrt{5}\right)\right)-\left(\left(-21\right)\sqrt{5}-\dfrac{85}{2}+\left(21\right)\sqrt{5}-\dfrac{355}{8}\right)\right)\\
&=&\dfrac{11}{2}+\dfrac{33}{4}+\left(\dfrac{93}{2}\right)\sqrt{5}-3+\dfrac{27}{2}-\dfrac{405}{8}+\left(\dfrac{36}{5}\right)\sqrt{5}+\left(0\right)\sqrt{5}+\dfrac{405}{8}+\left(\left(-\dfrac{260}{7}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{29}{3}\right)\sqrt{5}\right)+\dfrac{85}{2}-\left(\left(-\dfrac{66}{5}\right)\sqrt{5}\right)\right)-\left(\left(-21\right)\sqrt{5}-\dfrac{85}{2}+\left(21\right)\sqrt{5}-\dfrac{355}{8}\right)\\
&=&\dfrac{549}{8}+\left(-\dfrac{265}{42}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{11}{2}+\dfrac{33}{4}+\left(\dfrac{31}{2}\right)\sqrt{45}-3+\dfrac{27}{2}-\left(\left(\dfrac{81}{8}\right)\sqrt{25}\right)+\left(\dfrac{12}{5}\right)\sqrt{45}+\left(0\right)\sqrt{20}+\left(\dfrac{81}{8}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{52}{7}\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{29}{6}\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{33}{5}\right)\sqrt{20}\right)\right)-\left(\left(-7\right)\sqrt{45}+\left(-\dfrac{17}{2}\right)\sqrt{25}+\left(\dfrac{21}{5}\right)\sqrt{125}+\left(-\dfrac{71}{8}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{11}{2}+\dfrac{33}{4}+\left(\dfrac{93}{2}\right)\sqrt{5}-3+\dfrac{27}{2}-\dfrac{405}{8}+\left(\dfrac{36}{5}\right)\sqrt{5}+\left(0\right)\sqrt{5}+\dfrac{405}{8}\right)-\left(\left(\left(-\dfrac{260}{7}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{29}{3}\right)\sqrt{5}\right)+\dfrac{85}{2}-\left(\left(-\dfrac{66}{5}\right)\sqrt{5}\right)\right)-\left(\left(-21\right)\sqrt{5}-\dfrac{85}{2}+\left(21\right)\sqrt{5}-\dfrac{355}{8}\right)\right)\\
&=&\left(\dfrac{97}{4}+\left(\dfrac{537}{10}\right)\sqrt{5}\right)-\left(\left(-\dfrac{6301}{105}\right)\sqrt{5}+\dfrac{355}{8}\right)\\
&=&\dfrac{97}{4}+\left(\dfrac{537}{10}\right)\sqrt{5}+\left(\dfrac{6301}{105}\right)\sqrt{5}-\dfrac{355}{8}\\
&=&-\dfrac{161}{8}+\left(\dfrac{23879}{210}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{11}{2}+\dfrac{33}{4}+\left(\dfrac{31}{2}\right)\sqrt{45}-3+\dfrac{27}{2}-\left(\left(\dfrac{81}{8}\right)\sqrt{25}\right)+\left(\dfrac{12}{5}\right)\sqrt{45}+\left(0\right)\sqrt{20}+\left(\dfrac{81}{8}\right)\sqrt{25}\right)\times\left(\left(\left(-\dfrac{52}{7}\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{29}{6}\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{33}{5}\right)\sqrt{20}\right)\right)-\left(\left(-7\right)\sqrt{45}+\left(-\dfrac{17}{2}\right)\sqrt{25}+\left(\dfrac{21}{5}\right)\sqrt{125}+\left(-\dfrac{71}{8}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{11}{2}+\dfrac{33}{4}+\left(\dfrac{93}{2}\right)\sqrt{5}-3+\dfrac{27}{2}-\dfrac{405}{8}+\left(\dfrac{36}{5}\right)\sqrt{5}+\left(0\right)\sqrt{5}+\dfrac{405}{8}\right)\times\left(\left(\left(-\dfrac{260}{7}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{29}{3}\right)\sqrt{5}\right)+\dfrac{85}{2}-\left(\left(-\dfrac{66}{5}\right)\sqrt{5}\right)\right)-\left(\left(-21\right)\sqrt{5}-\dfrac{85}{2}+\left(21\right)\sqrt{5}-\dfrac{355}{8}\right)\right)\\
&=&\left(\dfrac{97}{4}+\left(\dfrac{537}{10}\right)\sqrt{5}\right)\left(\left(-\dfrac{6301}{105}\right)\sqrt{5}+\dfrac{355}{8}\right)\\
&=&\left(\dfrac{1558547}{1680}\right)\sqrt{5}+\dfrac{34435}{32}+\left(-\dfrac{1127879}{350}\right)\sqrt{25}\\
&=&\left(\dfrac{1558547}{1680}\right)\sqrt{5}-\dfrac{16840839}{1120}\\
\end{eqnarray*}