L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{3}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{38}{9}\right)\sqrt{125}\right)+\left(6\right)\sqrt{20}+\left(5\right)\sqrt{125}+\left(-\dfrac{56}{5}\right)\sqrt{45}\) et \( Y=\left(\left(-\dfrac{21}{2}\right)\sqrt{25}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}+\left(-8\right)\sqrt{25}+\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{125}\right)\) . 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(\dfrac{3}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{38}{9}\right)\sqrt{125}\right)+\left(6\right)\sqrt{20}+\left(5\right)\sqrt{125}+\left(-\dfrac{56}{5}\right)\sqrt{45}\right)+\left(\left(\left(-\dfrac{21}{2}\right)\sqrt{25}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}+\left(-8\right)\sqrt{25}+\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{190}{9}\right)\sqrt{5}\right)+\left(12\right)\sqrt{5}+\left(25\right)\sqrt{5}+\left(-\dfrac{168}{5}\right)\sqrt{5}\right)+\left(-\dfrac{105}{2}-\left(\left(27\right)\sqrt{5}-40+\left(27\right)\sqrt{5}\right)-\left(\left(\dfrac{165}{4}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{190}{9}\right)\sqrt{5}\right)+\left(12\right)\sqrt{5}+\left(25\right)\sqrt{5}+\left(-\dfrac{168}{5}\right)\sqrt{5}-\dfrac{105}{2}-\left(\left(27\right)\sqrt{5}-40+\left(27\right)\sqrt{5}\right)-\left(\left(\dfrac{165}{4}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{11383}{180}\right)\sqrt{5}-\dfrac{25}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{3}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{38}{9}\right)\sqrt{125}\right)+\left(6\right)\sqrt{20}+\left(5\right)\sqrt{125}+\left(-\dfrac{56}{5}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{21}{2}\right)\sqrt{25}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}+\left(-8\right)\sqrt{25}+\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{190}{9}\right)\sqrt{5}\right)+\left(12\right)\sqrt{5}+\left(25\right)\sqrt{5}+\left(-\dfrac{168}{5}\right)\sqrt{5}\right)-\left(-\dfrac{105}{2}-\left(\left(27\right)\sqrt{5}-40+\left(27\right)\sqrt{5}\right)-\left(\left(\dfrac{165}{4}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{2881}{90}\right)\sqrt{5}\right)-\left(-\dfrac{25}{2}+\left(-\dfrac{381}{4}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{2881}{90}\right)\sqrt{5}+\dfrac{25}{2}+\left(\dfrac{381}{4}\right)\sqrt{5}\\
&=&\left(\dfrac{22907}{180}\right)\sqrt{5}+\dfrac{25}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{3}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{38}{9}\right)\sqrt{125}\right)+\left(6\right)\sqrt{20}+\left(5\right)\sqrt{125}+\left(-\dfrac{56}{5}\right)\sqrt{45}\right)\times\left(\left(\left(-\dfrac{21}{2}\right)\sqrt{25}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}+\left(-8\right)\sqrt{25}+\left(\dfrac{27}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{190}{9}\right)\sqrt{5}\right)+\left(12\right)\sqrt{5}+\left(25\right)\sqrt{5}+\left(-\dfrac{168}{5}\right)\sqrt{5}\right)\times\left(-\dfrac{105}{2}-\left(\left(27\right)\sqrt{5}-40+\left(27\right)\sqrt{5}\right)-\left(\left(\dfrac{165}{4}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{2881}{90}\right)\sqrt{5}\right)\left(-\dfrac{25}{2}+\left(-\dfrac{381}{4}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{14405}{36}\right)\sqrt{5}+\left(-\dfrac{365887}{120}\right)\sqrt{25}\\
&=&\left(-\dfrac{14405}{36}\right)\sqrt{5}-\dfrac{365887}{24}\\
\end{eqnarray*}