L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-2\right)\sqrt{45}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)-\left(\left(-8\right)\sqrt{25}+\left(\dfrac{31}{9}\right)\sqrt{125}\right)\) et \( Y=\left(\dfrac{54}{5}\right)\sqrt{20}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-2\right)\sqrt{45}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)-\left(\left(-8\right)\sqrt{25}+\left(\dfrac{31}{9}\right)\sqrt{125}\right)\right)+\left(\left(\dfrac{54}{5}\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)-\left(-40+\left(\dfrac{155}{9}\right)\sqrt{5}\right)\right)+\left(\left(\dfrac{108}{5}\right)\sqrt{5}\right)\\
&=&\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)-\left(-40+\left(\dfrac{155}{9}\right)\sqrt{5}\right)+\left(\dfrac{108}{5}\right)\sqrt{5}\\
&=&\left(-\dfrac{551}{90}\right)\sqrt{5}+40\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-2\right)\sqrt{45}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)-\left(\left(-8\right)\sqrt{25}+\left(\dfrac{31}{9}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{54}{5}\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)-\left(-40+\left(\dfrac{155}{9}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{108}{5}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{499}{18}\right)\sqrt{5}+40\right)-\left(\left(\dfrac{108}{5}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{499}{18}\right)\sqrt{5}+40+\left(-\dfrac{108}{5}\right)\sqrt{5}\\
&=&\left(-\dfrac{4439}{90}\right)\sqrt{5}+40\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-2\right)\sqrt{45}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)-\left(\left(-8\right)\sqrt{25}+\left(\dfrac{31}{9}\right)\sqrt{125}\right)\right)\times\left(\left(\dfrac{54}{5}\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)-\left(-40+\left(\dfrac{155}{9}\right)\sqrt{5}\right)\right)\times\left(\left(\dfrac{108}{5}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{499}{18}\right)\sqrt{5}+40\right)\left(\left(\dfrac{108}{5}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{2994}{5}\right)\sqrt{25}+\left(864\right)\sqrt{5}\\
&=&-2994+\left(864\right)\sqrt{5}\\
\end{eqnarray*}