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 \( X=\left(\dfrac{22}{3}\right)\sqrt{125}\) et \( Y=\left(\left(\left(-\dfrac{31}{4}\right)\sqrt{25}\right)+\dfrac{53}{6}-\left(\left(0\right)\sqrt{20}\right)\right)+\dfrac{65}{2}-\left(\left(\left(-\dfrac{11}{3}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{20}\right)-\left(\left(\dfrac{71}{4}\right)\sqrt{125}\right)\right)\) . 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{22}{3}\right)\sqrt{125}\right)+\left(\left(\left(\left(-\dfrac{31}{4}\right)\sqrt{25}\right)+\dfrac{53}{6}-\left(\left(0\right)\sqrt{20}\right)\right)+\dfrac{65}{2}-\left(\left(\left(-\dfrac{11}{3}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{20}\right)-\left(\left(\dfrac{71}{4}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\dfrac{110}{3}\right)\sqrt{5}\right)+\left(\left(-\dfrac{155}{4}+\dfrac{53}{6}-\left(\left(0\right)\sqrt{5}\right)\right)+\dfrac{65}{2}-\left(\left(\left(-\dfrac{22}{3}\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)-\left(\left(\dfrac{355}{4}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\dfrac{110}{3}\right)\sqrt{5}+\left(-\dfrac{155}{4}+\dfrac{53}{6}-\left(\left(0\right)\sqrt{5}\right)\right)+\dfrac{65}{2}-\left(\left(\left(-\dfrac{22}{3}\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)-\left(\left(\dfrac{355}{4}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{563}{4}\right)\sqrt{5}+\dfrac{31}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{22}{3}\right)\sqrt{125}\right)-\left(\left(\left(\left(-\dfrac{31}{4}\right)\sqrt{25}\right)+\dfrac{53}{6}-\left(\left(0\right)\sqrt{20}\right)\right)+\dfrac{65}{2}-\left(\left(\left(-\dfrac{11}{3}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{20}\right)-\left(\left(\dfrac{71}{4}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\dfrac{110}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{155}{4}+\dfrac{53}{6}-\left(\left(0\right)\sqrt{5}\right)\right)+\dfrac{65}{2}-\left(\left(\left(-\dfrac{22}{3}\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)-\left(\left(\dfrac{355}{4}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(\dfrac{110}{3}\right)\sqrt{5}\right)-\left(\dfrac{31}{12}+\left(\dfrac{1249}{12}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{110}{3}\right)\sqrt{5}+-\dfrac{31}{12}+\left(-\dfrac{1249}{12}\right)\sqrt{5}\\
&=&\left(-\dfrac{809}{12}\right)\sqrt{5}-\dfrac{31}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{22}{3}\right)\sqrt{125}\right)\times\left(\left(\left(\left(-\dfrac{31}{4}\right)\sqrt{25}\right)+\dfrac{53}{6}-\left(\left(0\right)\sqrt{20}\right)\right)+\dfrac{65}{2}-\left(\left(\left(-\dfrac{11}{3}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{20}\right)-\left(\left(\dfrac{71}{4}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\dfrac{110}{3}\right)\sqrt{5}\right)\times\left(\left(-\dfrac{155}{4}+\dfrac{53}{6}-\left(\left(0\right)\sqrt{5}\right)\right)+\dfrac{65}{2}-\left(\left(\left(-\dfrac{22}{3}\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)-\left(\left(\dfrac{355}{4}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(\dfrac{110}{3}\right)\sqrt{5}\right)\left(\dfrac{31}{12}+\left(\dfrac{1249}{12}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{1705}{18}\right)\sqrt{5}+\left(\dfrac{68695}{18}\right)\sqrt{25}\\
&=&\left(\dfrac{1705}{18}\right)\sqrt{5}+\dfrac{343475}{18}\\
\end{eqnarray*}