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 \( X=\dfrac{61}{3}\) et \( Y=\left(\dfrac{61}{9}\right)\sqrt{125}+\left(\left(\dfrac{29}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{41}{3}\right)\sqrt{25}\right)-6+\dfrac{62}{5}-\left(\left(-\dfrac{17}{4}\right)\sqrt{45}\right)-\dfrac{23}{2}-\left(\left(-5\right)\sqrt{20}\right)+\left(\dfrac{9}{4}\right)\sqrt{20}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{61}{3}\right)+\left(\left(\dfrac{61}{9}\right)\sqrt{125}+\left(\left(\dfrac{29}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{41}{3}\right)\sqrt{25}\right)-6+\dfrac{62}{5}-\left(\left(-\dfrac{17}{4}\right)\sqrt{45}\right)-\dfrac{23}{2}-\left(\left(-5\right)\sqrt{20}\right)+\left(\dfrac{9}{4}\right)\sqrt{20}\right)\\
&=&\left(\dfrac{61}{3}\right)+\left(\left(\dfrac{305}{9}\right)\sqrt{5}+\dfrac{145}{4}-\dfrac{205}{3}-6+\dfrac{62}{5}-\left(\left(-\dfrac{51}{4}\right)\sqrt{5}\right)-\dfrac{23}{2}-\left(\left(-10\right)\sqrt{5}\right)+\left(\dfrac{9}{2}\right)\sqrt{5}\right)\\
&=&\dfrac{61}{3}+\left(\dfrac{305}{9}\right)\sqrt{5}+\dfrac{145}{4}-\dfrac{205}{3}-6+\dfrac{62}{5}-\left(\left(-\dfrac{51}{4}\right)\sqrt{5}\right)-\dfrac{23}{2}-\left(\left(-10\right)\sqrt{5}\right)+\left(\dfrac{9}{2}\right)\sqrt{5}\\
&=&-\dfrac{337}{20}+\left(\dfrac{2201}{36}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{61}{3}\right)-\left(\left(\dfrac{61}{9}\right)\sqrt{125}+\left(\left(\dfrac{29}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{41}{3}\right)\sqrt{25}\right)-6+\dfrac{62}{5}-\left(\left(-\dfrac{17}{4}\right)\sqrt{45}\right)-\dfrac{23}{2}-\left(\left(-5\right)\sqrt{20}\right)+\left(\dfrac{9}{4}\right)\sqrt{20}\right)\\
&=&\left(\dfrac{61}{3}\right)-\left(\left(\dfrac{305}{9}\right)\sqrt{5}+\dfrac{145}{4}-\dfrac{205}{3}-6+\dfrac{62}{5}-\left(\left(-\dfrac{51}{4}\right)\sqrt{5}\right)-\dfrac{23}{2}-\left(\left(-10\right)\sqrt{5}\right)+\left(\dfrac{9}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{61}{3}\right)-\left(\left(\dfrac{2201}{36}\right)\sqrt{5}-\dfrac{2231}{60}\right)\\
&=&\dfrac{61}{3}+\left(-\dfrac{2201}{36}\right)\sqrt{5}+\dfrac{2231}{60}\\
&=&\dfrac{3451}{60}+\left(-\dfrac{2201}{36}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{61}{3}\right)\times\left(\left(\dfrac{61}{9}\right)\sqrt{125}+\left(\left(\dfrac{29}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{41}{3}\right)\sqrt{25}\right)-6+\dfrac{62}{5}-\left(\left(-\dfrac{17}{4}\right)\sqrt{45}\right)-\dfrac{23}{2}-\left(\left(-5\right)\sqrt{20}\right)+\left(\dfrac{9}{4}\right)\sqrt{20}\right)\\
&=&\left(\dfrac{61}{3}\right)\times\left(\left(\dfrac{305}{9}\right)\sqrt{5}+\dfrac{145}{4}-\dfrac{205}{3}-6+\dfrac{62}{5}-\left(\left(-\dfrac{51}{4}\right)\sqrt{5}\right)-\dfrac{23}{2}-\left(\left(-10\right)\sqrt{5}\right)+\left(\dfrac{9}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{61}{3}\right)\left(\left(\dfrac{2201}{36}\right)\sqrt{5}-\dfrac{2231}{60}\right)\\
&=&\left(\dfrac{134261}{108}\right)\sqrt{5}-\dfrac{136091}{180}\\
&=&\left(\dfrac{134261}{108}\right)\sqrt{5}-\dfrac{136091}{180}\\
\end{eqnarray*}