L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{55}{9}\right)\sqrt{20}\) et \( Y=\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{125}\right)-8\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{25}+\left(\dfrac{36}{5}\right)\sqrt{20}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{55}{9}\right)\sqrt{20}\right)+\left(\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{125}\right)-8\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{25}+\left(\dfrac{36}{5}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\dfrac{110}{9}\right)\sqrt{5}\right)+\left(\left(\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-8\right)-\left(\dfrac{215}{9}+\left(\dfrac{72}{5}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{110}{9}\right)\sqrt{5}+\left(\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-8\right)-\left(\dfrac{215}{9}+\left(\dfrac{72}{5}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{2221}{90}\right)\sqrt{5}-\dfrac{287}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{55}{9}\right)\sqrt{20}\right)-\left(\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{125}\right)-8\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{25}+\left(\dfrac{36}{5}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\dfrac{110}{9}\right)\sqrt{5}\right)-\left(\left(\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-8\right)-\left(\dfrac{215}{9}+\left(\dfrac{72}{5}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{110}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{369}{10}\right)\sqrt{5}-\dfrac{287}{9}\right)\\
&=&\left(\dfrac{110}{9}\right)\sqrt{5}+\left(\dfrac{369}{10}\right)\sqrt{5}+\dfrac{287}{9}\\
&=&\left(\dfrac{4421}{90}\right)\sqrt{5}+\dfrac{287}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{55}{9}\right)\sqrt{20}\right)\times\left(\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{125}\right)-8\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{25}+\left(\dfrac{36}{5}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\dfrac{110}{9}\right)\sqrt{5}\right)\times\left(\left(\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)-8\right)-\left(\dfrac{215}{9}+\left(\dfrac{72}{5}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{110}{9}\right)\sqrt{5}\right)\left(\left(-\dfrac{369}{10}\right)\sqrt{5}-\dfrac{287}{9}\right)\\
&=&\left(-451\right)\sqrt{25}+\left(-\dfrac{31570}{81}\right)\sqrt{5}\\
&=&-2255+\left(-\dfrac{31570}{81}\right)\sqrt{5}\\
\end{eqnarray*}