L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{13}{2}\right)\sqrt{125}\right)-\left(\left(-9\right)\sqrt{125}\right)-\left(\left(4\right)\sqrt{20}\right)+\left(\dfrac{8}{9}\right)\sqrt{125}+\left(\dfrac{41}{9}\right)\sqrt{45}-8\) et \( Y=\left(\left(\dfrac{71}{8}\right)\sqrt{45}+\left(\dfrac{63}{4}\right)\sqrt{45}+\left(-\dfrac{56}{9}\right)\sqrt{125}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)\) . 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{13}{2}\right)\sqrt{125}\right)-\left(\left(-9\right)\sqrt{125}\right)-\left(\left(4\right)\sqrt{20}\right)+\left(\dfrac{8}{9}\right)\sqrt{125}+\left(\dfrac{41}{9}\right)\sqrt{45}-8\right)+\left(\left(\left(\dfrac{71}{8}\right)\sqrt{45}+\left(\dfrac{63}{4}\right)\sqrt{45}+\left(-\dfrac{56}{9}\right)\sqrt{125}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(-\dfrac{65}{2}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)+\left(\dfrac{40}{9}\right)\sqrt{5}+\left(\dfrac{41}{3}\right)\sqrt{5}-8\right)+\left(\left(\left(\dfrac{213}{8}\right)\sqrt{5}+\left(\dfrac{189}{4}\right)\sqrt{5}+\left(-\dfrac{280}{9}\right)\sqrt{5}\right)-\left(\left(27\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{65}{2}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)+\left(\dfrac{40}{9}\right)\sqrt{5}+\left(\dfrac{41}{3}\right)\sqrt{5}-8+\left(\left(\dfrac{213}{8}\right)\sqrt{5}+\left(\dfrac{189}{4}\right)\sqrt{5}+\left(-\dfrac{280}{9}\right)\sqrt{5}\right)-\left(\left(27\right)\sqrt{5}\right)\\
&=&\left(\dfrac{307}{8}\right)\sqrt{5}-8\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{13}{2}\right)\sqrt{125}\right)-\left(\left(-9\right)\sqrt{125}\right)-\left(\left(4\right)\sqrt{20}\right)+\left(\dfrac{8}{9}\right)\sqrt{125}+\left(\dfrac{41}{9}\right)\sqrt{45}-8\right)-\left(\left(\left(\dfrac{71}{8}\right)\sqrt{45}+\left(\dfrac{63}{4}\right)\sqrt{45}+\left(-\dfrac{56}{9}\right)\sqrt{125}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(-\dfrac{65}{2}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)+\left(\dfrac{40}{9}\right)\sqrt{5}+\left(\dfrac{41}{3}\right)\sqrt{5}-8\right)-\left(\left(\left(\dfrac{213}{8}\right)\sqrt{5}+\left(\dfrac{189}{4}\right)\sqrt{5}+\left(-\dfrac{280}{9}\right)\sqrt{5}\right)-\left(\left(27\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{407}{18}\right)\sqrt{5}-8\right)-\left(\left(\dfrac{1135}{72}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{407}{18}\right)\sqrt{5}-8+\left(-\dfrac{1135}{72}\right)\sqrt{5}\\
&=&\left(\dfrac{493}{72}\right)\sqrt{5}-8\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{13}{2}\right)\sqrt{125}\right)-\left(\left(-9\right)\sqrt{125}\right)-\left(\left(4\right)\sqrt{20}\right)+\left(\dfrac{8}{9}\right)\sqrt{125}+\left(\dfrac{41}{9}\right)\sqrt{45}-8\right)\times\left(\left(\left(\dfrac{71}{8}\right)\sqrt{45}+\left(\dfrac{63}{4}\right)\sqrt{45}+\left(-\dfrac{56}{9}\right)\sqrt{125}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(-\dfrac{65}{2}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)+\left(\dfrac{40}{9}\right)\sqrt{5}+\left(\dfrac{41}{3}\right)\sqrt{5}-8\right)\times\left(\left(\left(\dfrac{213}{8}\right)\sqrt{5}+\left(\dfrac{189}{4}\right)\sqrt{5}+\left(-\dfrac{280}{9}\right)\sqrt{5}\right)-\left(\left(27\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{407}{18}\right)\sqrt{5}-8\right)\left(\left(\dfrac{1135}{72}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{461945}{1296}\right)\sqrt{25}+\left(-\dfrac{1135}{9}\right)\sqrt{5}\\
&=&\dfrac{2309725}{1296}+\left(-\dfrac{1135}{9}\right)\sqrt{5}\\
\end{eqnarray*}