L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(0\right)\sqrt{45}+\left(-\dfrac{23}{9}\right)\sqrt{125}+\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(-\dfrac{11}{3}\right)\sqrt{25}\) et \( Y=0+\left(\left(-\dfrac{23}{6}\right)\sqrt{45}\right)+\dfrac{16}{3}+\dfrac{25}{4}+\left(-2\right)\sqrt{125}+\left(\dfrac{1}{9}\right)\sqrt{45}+\left(\dfrac{38}{9}\right)\sqrt{125}+\left(\left(-\dfrac{29}{6}\right)\sqrt{20}\right)-3\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(0\right)\sqrt{45}+\left(-\dfrac{23}{9}\right)\sqrt{125}+\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(-\dfrac{11}{3}\right)\sqrt{25}\right)+\left(0+\left(\left(-\dfrac{23}{6}\right)\sqrt{45}\right)+\dfrac{16}{3}+\dfrac{25}{4}+\left(-2\right)\sqrt{125}+\left(\dfrac{1}{9}\right)\sqrt{45}+\left(\dfrac{38}{9}\right)\sqrt{125}+\left(\left(-\dfrac{29}{6}\right)\sqrt{20}\right)-3\right)\\
&=&\left(\left(0\right)\sqrt{5}+\left(-\dfrac{115}{9}\right)\sqrt{5}-\dfrac{115}{3}-\dfrac{55}{3}\right)+\left(0+\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)+\dfrac{16}{3}+\dfrac{25}{4}+\left(-10\right)\sqrt{5}+\left(\dfrac{1}{3}\right)\sqrt{5}+\left(\dfrac{190}{9}\right)\sqrt{5}+\left(\left(-\dfrac{29}{3}\right)\sqrt{5}\right)-3\right)\\
&=&\left(0\right)\sqrt{5}+\left(-\dfrac{115}{9}\right)\sqrt{5}-\dfrac{115}{3}-\dfrac{55}{3}+0+\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)+\dfrac{16}{3}+\dfrac{25}{4}+\left(-10\right)\sqrt{5}+\left(\dfrac{1}{3}\right)\sqrt{5}+\left(\dfrac{190}{9}\right)\sqrt{5}+\left(\left(-\dfrac{29}{3}\right)\sqrt{5}\right)-3\\
&=&\left(-\dfrac{45}{2}\right)\sqrt{5}-\dfrac{577}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(0\right)\sqrt{45}+\left(-\dfrac{23}{9}\right)\sqrt{125}+\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(-\dfrac{11}{3}\right)\sqrt{25}\right)-\left(0+\left(\left(-\dfrac{23}{6}\right)\sqrt{45}\right)+\dfrac{16}{3}+\dfrac{25}{4}+\left(-2\right)\sqrt{125}+\left(\dfrac{1}{9}\right)\sqrt{45}+\left(\dfrac{38}{9}\right)\sqrt{125}+\left(\left(-\dfrac{29}{6}\right)\sqrt{20}\right)-3\right)\\
&=&\left(\left(0\right)\sqrt{5}+\left(-\dfrac{115}{9}\right)\sqrt{5}-\dfrac{115}{3}-\dfrac{55}{3}\right)-\left(0+\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)+\dfrac{16}{3}+\dfrac{25}{4}+\left(-10\right)\sqrt{5}+\left(\dfrac{1}{3}\right)\sqrt{5}+\left(\dfrac{190}{9}\right)\sqrt{5}+\left(\left(-\dfrac{29}{3}\right)\sqrt{5}\right)-3\right)\\
&=&\left(\left(-\dfrac{115}{9}\right)\sqrt{5}-\dfrac{170}{3}\right)-\left(\dfrac{103}{12}+\left(-\dfrac{175}{18}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{115}{9}\right)\sqrt{5}-\dfrac{170}{3}+-\dfrac{103}{12}+\left(\dfrac{175}{18}\right)\sqrt{5}\\
&=&\left(-\dfrac{55}{18}\right)\sqrt{5}-\dfrac{261}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(0\right)\sqrt{45}+\left(-\dfrac{23}{9}\right)\sqrt{125}+\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(-\dfrac{11}{3}\right)\sqrt{25}\right)\times\left(0+\left(\left(-\dfrac{23}{6}\right)\sqrt{45}\right)+\dfrac{16}{3}+\dfrac{25}{4}+\left(-2\right)\sqrt{125}+\left(\dfrac{1}{9}\right)\sqrt{45}+\left(\dfrac{38}{9}\right)\sqrt{125}+\left(\left(-\dfrac{29}{6}\right)\sqrt{20}\right)-3\right)\\
&=&\left(\left(0\right)\sqrt{5}+\left(-\dfrac{115}{9}\right)\sqrt{5}-\dfrac{115}{3}-\dfrac{55}{3}\right)\times\left(0+\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)+\dfrac{16}{3}+\dfrac{25}{4}+\left(-10\right)\sqrt{5}+\left(\dfrac{1}{3}\right)\sqrt{5}+\left(\dfrac{190}{9}\right)\sqrt{5}+\left(\left(-\dfrac{29}{3}\right)\sqrt{5}\right)-3\right)\\
&=&\left(\left(-\dfrac{115}{9}\right)\sqrt{5}-\dfrac{170}{3}\right)\left(\dfrac{103}{12}+\left(-\dfrac{175}{18}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{1765}{4}\right)\sqrt{5}+\left(\dfrac{20125}{162}\right)\sqrt{25}-\dfrac{8755}{18}\\
&=&\left(\dfrac{1765}{4}\right)\sqrt{5}+\dfrac{10915}{81}\\
\end{eqnarray*}