L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{63}{4}\right)\sqrt{125}+\left(\dfrac{65}{8}\right)\sqrt{125}+\left(2\right)\sqrt{45}-\dfrac{12}{7}+\left(8\right)\sqrt{125}+\left(-\dfrac{11}{2}\right)\sqrt{45}+\left(\dfrac{61}{7}\right)\sqrt{20}-\dfrac{79}{7}+\left(-\dfrac{3}{7}\right)\sqrt{20}+\left(-\dfrac{47}{2}\right)\sqrt{20}+\left(1\right)\sqrt{20}\) et \( Y=\left(-4\right)\sqrt{25}+\left(\dfrac{16}{3}\right)\sqrt{25}+\left(\dfrac{11}{2}\right)\sqrt{45}+\left(\dfrac{27}{8}\right)\sqrt{125}\) . 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{63}{4}\right)\sqrt{125}+\left(\dfrac{65}{8}\right)\sqrt{125}+\left(2\right)\sqrt{45}-\dfrac{12}{7}+\left(8\right)\sqrt{125}+\left(-\dfrac{11}{2}\right)\sqrt{45}+\left(\dfrac{61}{7}\right)\sqrt{20}-\dfrac{79}{7}+\left(-\dfrac{3}{7}\right)\sqrt{20}+\left(-\dfrac{47}{2}\right)\sqrt{20}+\left(1\right)\sqrt{20}\right)+\left(\left(-4\right)\sqrt{25}+\left(\dfrac{16}{3}\right)\sqrt{25}+\left(\dfrac{11}{2}\right)\sqrt{45}+\left(\dfrac{27}{8}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{315}{4}\right)\sqrt{5}+\left(\dfrac{325}{8}\right)\sqrt{5}+\left(6\right)\sqrt{5}-\dfrac{12}{7}+\left(40\right)\sqrt{5}+\left(-\dfrac{33}{2}\right)\sqrt{5}+\left(\dfrac{122}{7}\right)\sqrt{5}-\dfrac{79}{7}+\left(-\dfrac{6}{7}\right)\sqrt{5}+\left(-47\right)\sqrt{5}+\left(2\right)\sqrt{5}\right)+\left(-20+\dfrac{80}{3}+\left(\dfrac{33}{2}\right)\sqrt{5}+\left(\dfrac{135}{8}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{315}{4}\right)\sqrt{5}+\left(\dfrac{325}{8}\right)\sqrt{5}+\left(6\right)\sqrt{5}-\dfrac{12}{7}+\left(40\right)\sqrt{5}+\left(-\dfrac{33}{2}\right)\sqrt{5}+\left(\dfrac{122}{7}\right)\sqrt{5}-\dfrac{79}{7}+\left(-\dfrac{6}{7}\right)\sqrt{5}+\left(-47\right)\sqrt{5}+\left(2\right)\sqrt{5}-20+\dfrac{80}{3}+\left(\dfrac{33}{2}\right)\sqrt{5}+\left(\dfrac{135}{8}\right)\sqrt{5}\\
&=&\left(\dfrac{4307}{28}\right)\sqrt{5}-\dfrac{19}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{63}{4}\right)\sqrt{125}+\left(\dfrac{65}{8}\right)\sqrt{125}+\left(2\right)\sqrt{45}-\dfrac{12}{7}+\left(8\right)\sqrt{125}+\left(-\dfrac{11}{2}\right)\sqrt{45}+\left(\dfrac{61}{7}\right)\sqrt{20}-\dfrac{79}{7}+\left(-\dfrac{3}{7}\right)\sqrt{20}+\left(-\dfrac{47}{2}\right)\sqrt{20}+\left(1\right)\sqrt{20}\right)-\left(\left(-4\right)\sqrt{25}+\left(\dfrac{16}{3}\right)\sqrt{25}+\left(\dfrac{11}{2}\right)\sqrt{45}+\left(\dfrac{27}{8}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{315}{4}\right)\sqrt{5}+\left(\dfrac{325}{8}\right)\sqrt{5}+\left(6\right)\sqrt{5}-\dfrac{12}{7}+\left(40\right)\sqrt{5}+\left(-\dfrac{33}{2}\right)\sqrt{5}+\left(\dfrac{122}{7}\right)\sqrt{5}-\dfrac{79}{7}+\left(-\dfrac{6}{7}\right)\sqrt{5}+\left(-47\right)\sqrt{5}+\left(2\right)\sqrt{5}\right)-\left(-20+\dfrac{80}{3}+\left(\dfrac{33}{2}\right)\sqrt{5}+\left(\dfrac{135}{8}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{6745}{56}\right)\sqrt{5}-13\right)-\left(\dfrac{20}{3}+\left(\dfrac{267}{8}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{6745}{56}\right)\sqrt{5}-13+-\dfrac{20}{3}+\left(-\dfrac{267}{8}\right)\sqrt{5}\\
&=&\left(\dfrac{1219}{14}\right)\sqrt{5}-\dfrac{59}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{63}{4}\right)\sqrt{125}+\left(\dfrac{65}{8}\right)\sqrt{125}+\left(2\right)\sqrt{45}-\dfrac{12}{7}+\left(8\right)\sqrt{125}+\left(-\dfrac{11}{2}\right)\sqrt{45}+\left(\dfrac{61}{7}\right)\sqrt{20}-\dfrac{79}{7}+\left(-\dfrac{3}{7}\right)\sqrt{20}+\left(-\dfrac{47}{2}\right)\sqrt{20}+\left(1\right)\sqrt{20}\right)\times\left(\left(-4\right)\sqrt{25}+\left(\dfrac{16}{3}\right)\sqrt{25}+\left(\dfrac{11}{2}\right)\sqrt{45}+\left(\dfrac{27}{8}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{315}{4}\right)\sqrt{5}+\left(\dfrac{325}{8}\right)\sqrt{5}+\left(6\right)\sqrt{5}-\dfrac{12}{7}+\left(40\right)\sqrt{5}+\left(-\dfrac{33}{2}\right)\sqrt{5}+\left(\dfrac{122}{7}\right)\sqrt{5}-\dfrac{79}{7}+\left(-\dfrac{6}{7}\right)\sqrt{5}+\left(-47\right)\sqrt{5}+\left(2\right)\sqrt{5}\right)\times\left(-20+\dfrac{80}{3}+\left(\dfrac{33}{2}\right)\sqrt{5}+\left(\dfrac{135}{8}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{6745}{56}\right)\sqrt{5}-13\right)\left(\dfrac{20}{3}+\left(\dfrac{267}{8}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{62009}{168}\right)\sqrt{5}+\left(\dfrac{1800915}{448}\right)\sqrt{25}-\dfrac{260}{3}\\
&=&\left(\dfrac{62009}{168}\right)\sqrt{5}+\dfrac{26897245}{1344}\\
\end{eqnarray*}