L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(6\right)\sqrt{125}+\left(-3\right)\sqrt{20}+\left(\dfrac{17}{6}\right)\sqrt{45}+5\right)-\left(\left(-5\right)\sqrt{25}+\left(\dfrac{25}{7}\right)\sqrt{20}+\left(-\dfrac{24}{5}\right)\sqrt{125}\right)-\dfrac{21}{2}\) et \( Y=\left(\left(\dfrac{26}{3}\right)\sqrt{25}+\left(\dfrac{67}{8}\right)\sqrt{125}\right)-\left(\left(0\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{20}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{20}\right)\right)\) . 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(6\right)\sqrt{125}+\left(-3\right)\sqrt{20}+\left(\dfrac{17}{6}\right)\sqrt{45}+5\right)-\left(\left(-5\right)\sqrt{25}+\left(\dfrac{25}{7}\right)\sqrt{20}+\left(-\dfrac{24}{5}\right)\sqrt{125}\right)-\dfrac{21}{2}\right)+\left(\left(\left(\dfrac{26}{3}\right)\sqrt{25}+\left(\dfrac{67}{8}\right)\sqrt{125}\right)-\left(\left(0\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{20}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{20}\right)\right)\right)\\
&=&\left(\left(\left(30\right)\sqrt{5}+\left(-6\right)\sqrt{5}+\left(\dfrac{17}{2}\right)\sqrt{5}+5\right)-\left(-25+\left(\dfrac{50}{7}\right)\sqrt{5}+\left(-24\right)\sqrt{5}\right)-\dfrac{21}{2}\right)+\left(\left(\dfrac{130}{3}+\left(\dfrac{335}{8}\right)\sqrt{5}\right)-\left(\left(0\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{104}{5}\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(30\right)\sqrt{5}+\left(-6\right)\sqrt{5}+\left(\dfrac{17}{2}\right)\sqrt{5}+5\right)-\left(-25+\left(\dfrac{50}{7}\right)\sqrt{5}+\left(-24\right)\sqrt{5}\right)-\dfrac{21}{2}+\left(\dfrac{130}{3}+\left(\dfrac{335}{8}\right)\sqrt{5}\right)-\left(\left(0\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{104}{5}\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{73387}{840}\right)\sqrt{5}+\dfrac{377}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(6\right)\sqrt{125}+\left(-3\right)\sqrt{20}+\left(\dfrac{17}{6}\right)\sqrt{45}+5\right)-\left(\left(-5\right)\sqrt{25}+\left(\dfrac{25}{7}\right)\sqrt{20}+\left(-\dfrac{24}{5}\right)\sqrt{125}\right)-\dfrac{21}{2}\right)-\left(\left(\left(\dfrac{26}{3}\right)\sqrt{25}+\left(\dfrac{67}{8}\right)\sqrt{125}\right)-\left(\left(0\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{20}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{20}\right)\right)\right)\\
&=&\left(\left(\left(30\right)\sqrt{5}+\left(-6\right)\sqrt{5}+\left(\dfrac{17}{2}\right)\sqrt{5}+5\right)-\left(-25+\left(\dfrac{50}{7}\right)\sqrt{5}+\left(-24\right)\sqrt{5}\right)-\dfrac{21}{2}\right)-\left(\left(\dfrac{130}{3}+\left(\dfrac{335}{8}\right)\sqrt{5}\right)-\left(\left(0\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{104}{5}\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(\dfrac{691}{14}\right)\sqrt{5}+\dfrac{39}{2}\right)-\left(\dfrac{130}{3}+\left(\dfrac{4561}{120}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{691}{14}\right)\sqrt{5}+\dfrac{39}{2}+-\dfrac{130}{3}+\left(-\dfrac{4561}{120}\right)\sqrt{5}\\
&=&\left(\dfrac{9533}{840}\right)\sqrt{5}-\dfrac{143}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(6\right)\sqrt{125}+\left(-3\right)\sqrt{20}+\left(\dfrac{17}{6}\right)\sqrt{45}+5\right)-\left(\left(-5\right)\sqrt{25}+\left(\dfrac{25}{7}\right)\sqrt{20}+\left(-\dfrac{24}{5}\right)\sqrt{125}\right)-\dfrac{21}{2}\right)\times\left(\left(\left(\dfrac{26}{3}\right)\sqrt{25}+\left(\dfrac{67}{8}\right)\sqrt{125}\right)-\left(\left(0\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{20}\right)-\left(\left(-\dfrac{20}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{20}\right)\right)\right)\\
&=&\left(\left(\left(30\right)\sqrt{5}+\left(-6\right)\sqrt{5}+\left(\dfrac{17}{2}\right)\sqrt{5}+5\right)-\left(-25+\left(\dfrac{50}{7}\right)\sqrt{5}+\left(-24\right)\sqrt{5}\right)-\dfrac{21}{2}\right)\times\left(\left(\dfrac{130}{3}+\left(\dfrac{335}{8}\right)\sqrt{5}\right)-\left(\left(0\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{104}{5}\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(\dfrac{691}{14}\right)\sqrt{5}+\dfrac{39}{2}\right)\left(\dfrac{130}{3}+\left(\dfrac{4561}{120}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{4838353}{1680}\right)\sqrt{5}+\left(\dfrac{3151651}{1680}\right)\sqrt{25}+845\\
&=&\left(\dfrac{4838353}{1680}\right)\sqrt{5}+\dfrac{3435571}{336}\\
\end{eqnarray*}