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 \( X=\left(\dfrac{31}{3}\right)\sqrt{45}+\dfrac{29}{7}-6\) et \( Y=\left(\left(\dfrac{68}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{32}{5}\right)\sqrt{45}+\left(-\dfrac{3}{7}\right)\sqrt{25}\right)\) . 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{31}{3}\right)\sqrt{45}+\dfrac{29}{7}-6\right)+\left(\left(\left(\dfrac{68}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{32}{5}\right)\sqrt{45}+\left(-\dfrac{3}{7}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(31\right)\sqrt{5}+\dfrac{29}{7}-6\right)+\left(\dfrac{340}{9}-\left(\left(\dfrac{96}{5}\right)\sqrt{5}-\dfrac{15}{7}\right)\right)\\
&=&\left(31\right)\sqrt{5}+\dfrac{29}{7}-6+\dfrac{340}{9}-\left(\left(\dfrac{96}{5}\right)\sqrt{5}-\dfrac{15}{7}\right)\\
&=&\left(\dfrac{59}{5}\right)\sqrt{5}+\dfrac{2398}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{31}{3}\right)\sqrt{45}+\dfrac{29}{7}-6\right)-\left(\left(\left(\dfrac{68}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{32}{5}\right)\sqrt{45}+\left(-\dfrac{3}{7}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(31\right)\sqrt{5}+\dfrac{29}{7}-6\right)-\left(\dfrac{340}{9}-\left(\left(\dfrac{96}{5}\right)\sqrt{5}-\dfrac{15}{7}\right)\right)\\
&=&\left(\left(31\right)\sqrt{5}-\dfrac{13}{7}\right)-\left(\dfrac{2515}{63}+\left(-\dfrac{96}{5}\right)\sqrt{5}\right)\\
&=&\left(31\right)\sqrt{5}-\dfrac{13}{7}+-\dfrac{2515}{63}+\left(\dfrac{96}{5}\right)\sqrt{5}\\
&=&\left(\dfrac{251}{5}\right)\sqrt{5}-\dfrac{376}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{31}{3}\right)\sqrt{45}+\dfrac{29}{7}-6\right)\times\left(\left(\left(\dfrac{68}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{32}{5}\right)\sqrt{45}+\left(-\dfrac{3}{7}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(31\right)\sqrt{5}+\dfrac{29}{7}-6\right)\times\left(\dfrac{340}{9}-\left(\left(\dfrac{96}{5}\right)\sqrt{5}-\dfrac{15}{7}\right)\right)\\
&=&\left(\left(31\right)\sqrt{5}-\dfrac{13}{7}\right)\left(\dfrac{2515}{63}+\left(-\dfrac{96}{5}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{401057}{315}\right)\sqrt{5}+\left(-\dfrac{2976}{5}\right)\sqrt{25}-\dfrac{32695}{441}\\
&=&\left(\dfrac{401057}{315}\right)\sqrt{5}-\dfrac{1345111}{441}\\
\end{eqnarray*}