L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{27}{4}\right)\sqrt{20}+\left(\dfrac{34}{9}\right)\sqrt{125}+\left(\dfrac{55}{6}\right)\sqrt{45}+\left(-\dfrac{13}{2}\right)\sqrt{45}+\left(\left(\dfrac{28}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{20}\right)\) et \( Y=\left(9\right)\sqrt{20}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{27}{4}\right)\sqrt{20}+\left(\dfrac{34}{9}\right)\sqrt{125}+\left(\dfrac{55}{6}\right)\sqrt{45}+\left(-\dfrac{13}{2}\right)\sqrt{45}+\left(\left(\dfrac{28}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{20}\right)\right)+\left(\left(9\right)\sqrt{20}\right)\\
&=&\left(\left(\dfrac{27}{2}\right)\sqrt{5}+\left(\dfrac{170}{9}\right)\sqrt{5}+\left(\dfrac{55}{2}\right)\sqrt{5}+\left(-\dfrac{39}{2}\right)\sqrt{5}+\left(\left(\dfrac{56}{5}\right)\sqrt{5}\right)-\left(\left(7\right)\sqrt{5}\right)\right)+\left(\left(18\right)\sqrt{5}\right)\\
&=&\left(\dfrac{27}{2}\right)\sqrt{5}+\left(\dfrac{170}{9}\right)\sqrt{5}+\left(\dfrac{55}{2}\right)\sqrt{5}+\left(-\dfrac{39}{2}\right)\sqrt{5}+\left(\left(\dfrac{56}{5}\right)\sqrt{5}\right)-\left(\left(7\right)\sqrt{5}\right)+\left(18\right)\sqrt{5}\\
&=&\left(\dfrac{5633}{90}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{27}{4}\right)\sqrt{20}+\left(\dfrac{34}{9}\right)\sqrt{125}+\left(\dfrac{55}{6}\right)\sqrt{45}+\left(-\dfrac{13}{2}\right)\sqrt{45}+\left(\left(\dfrac{28}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{20}\right)\right)-\left(\left(9\right)\sqrt{20}\right)\\
&=&\left(\left(\dfrac{27}{2}\right)\sqrt{5}+\left(\dfrac{170}{9}\right)\sqrt{5}+\left(\dfrac{55}{2}\right)\sqrt{5}+\left(-\dfrac{39}{2}\right)\sqrt{5}+\left(\left(\dfrac{56}{5}\right)\sqrt{5}\right)-\left(\left(7\right)\sqrt{5}\right)\right)-\left(\left(18\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{4013}{90}\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)\\
&=&\left(\dfrac{4013}{90}\right)\sqrt{5}+\left(-18\right)\sqrt{5}\\
&=&\left(\dfrac{2393}{90}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{27}{4}\right)\sqrt{20}+\left(\dfrac{34}{9}\right)\sqrt{125}+\left(\dfrac{55}{6}\right)\sqrt{45}+\left(-\dfrac{13}{2}\right)\sqrt{45}+\left(\left(\dfrac{28}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{20}\right)\right)\times\left(\left(9\right)\sqrt{20}\right)\\
&=&\left(\left(\dfrac{27}{2}\right)\sqrt{5}+\left(\dfrac{170}{9}\right)\sqrt{5}+\left(\dfrac{55}{2}\right)\sqrt{5}+\left(-\dfrac{39}{2}\right)\sqrt{5}+\left(\left(\dfrac{56}{5}\right)\sqrt{5}\right)-\left(\left(7\right)\sqrt{5}\right)\right)\times\left(\left(18\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{4013}{90}\right)\sqrt{5}\right)\left(\left(18\right)\sqrt{5}\right)\\
&=&\left(\dfrac{4013}{5}\right)\sqrt{25}\\
&=&4013\\
\end{eqnarray*}