L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(1\right)\sqrt{45}\) et \( Y=\left(\left(5\right)\sqrt{25}+\left(-\dfrac{27}{4}\right)\sqrt{45}+\left(\dfrac{25}{2}\right)\sqrt{25}-\dfrac{38}{3}+\left(-\dfrac{41}{9}\right)\sqrt{20}\right)-\left(\left(-\dfrac{7}{8}\right)\sqrt{125}+\left(\dfrac{9}{2}\right)\sqrt{45}+\left(-\dfrac{61}{6}\right)\sqrt{25}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(1\right)\sqrt{45}\right)+\left(\left(\left(5\right)\sqrt{25}+\left(-\dfrac{27}{4}\right)\sqrt{45}+\left(\dfrac{25}{2}\right)\sqrt{25}-\dfrac{38}{3}+\left(-\dfrac{41}{9}\right)\sqrt{20}\right)-\left(\left(-\dfrac{7}{8}\right)\sqrt{125}+\left(\dfrac{9}{2}\right)\sqrt{45}+\left(-\dfrac{61}{6}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(3\right)\sqrt{5}\right)+\left(\left(25+\left(-\dfrac{81}{4}\right)\sqrt{5}+\dfrac{125}{2}-\dfrac{38}{3}+\left(-\dfrac{82}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{5}+\left(\dfrac{27}{2}\right)\sqrt{5}-\dfrac{305}{6}\right)\right)\\
&=&\left(3\right)\sqrt{5}+\left(25+\left(-\dfrac{81}{4}\right)\sqrt{5}+\dfrac{125}{2}-\dfrac{38}{3}+\left(-\dfrac{82}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{5}+\left(\dfrac{27}{2}\right)\sqrt{5}-\dfrac{305}{6}\right)\\
&=&\left(-\dfrac{2555}{72}\right)\sqrt{5}+\dfrac{377}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(1\right)\sqrt{45}\right)-\left(\left(\left(5\right)\sqrt{25}+\left(-\dfrac{27}{4}\right)\sqrt{45}+\left(\dfrac{25}{2}\right)\sqrt{25}-\dfrac{38}{3}+\left(-\dfrac{41}{9}\right)\sqrt{20}\right)-\left(\left(-\dfrac{7}{8}\right)\sqrt{125}+\left(\dfrac{9}{2}\right)\sqrt{45}+\left(-\dfrac{61}{6}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(3\right)\sqrt{5}\right)-\left(\left(25+\left(-\dfrac{81}{4}\right)\sqrt{5}+\dfrac{125}{2}-\dfrac{38}{3}+\left(-\dfrac{82}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{5}+\left(\dfrac{27}{2}\right)\sqrt{5}-\dfrac{305}{6}\right)\right)\\
&=&\left(\left(3\right)\sqrt{5}\right)-\left(\dfrac{377}{3}+\left(-\dfrac{2771}{72}\right)\sqrt{5}\right)\\
&=&\left(3\right)\sqrt{5}+-\dfrac{377}{3}+\left(\dfrac{2771}{72}\right)\sqrt{5}\\
&=&\left(\dfrac{2987}{72}\right)\sqrt{5}-\dfrac{377}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(1\right)\sqrt{45}\right)\times\left(\left(\left(5\right)\sqrt{25}+\left(-\dfrac{27}{4}\right)\sqrt{45}+\left(\dfrac{25}{2}\right)\sqrt{25}-\dfrac{38}{3}+\left(-\dfrac{41}{9}\right)\sqrt{20}\right)-\left(\left(-\dfrac{7}{8}\right)\sqrt{125}+\left(\dfrac{9}{2}\right)\sqrt{45}+\left(-\dfrac{61}{6}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(3\right)\sqrt{5}\right)\times\left(\left(25+\left(-\dfrac{81}{4}\right)\sqrt{5}+\dfrac{125}{2}-\dfrac{38}{3}+\left(-\dfrac{82}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{5}+\left(\dfrac{27}{2}\right)\sqrt{5}-\dfrac{305}{6}\right)\right)\\
&=&\left(\left(3\right)\sqrt{5}\right)\left(\dfrac{377}{3}+\left(-\dfrac{2771}{72}\right)\sqrt{5}\right)\\
&=&\left(377\right)\sqrt{5}+\left(-\dfrac{2771}{24}\right)\sqrt{25}\\
&=&\left(377\right)\sqrt{5}-\dfrac{13855}{24}\\
\end{eqnarray*}