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 \( X=\left(\dfrac{11}{6}\right)\sqrt{45}\) et \( Y=\left(\left(\dfrac{73}{7}\right)\sqrt{20}+\left(-\dfrac{37}{4}\right)\sqrt{125}+\dfrac{1}{4}\right)-\left(\left(\dfrac{37}{5}\right)\sqrt{20}-\dfrac{4}{7}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}+\left(\dfrac{22}{3}\right)\sqrt{25}\right)\) . 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{11}{6}\right)\sqrt{45}\right)+\left(\left(\left(\dfrac{73}{7}\right)\sqrt{20}+\left(-\dfrac{37}{4}\right)\sqrt{125}+\dfrac{1}{4}\right)-\left(\left(\dfrac{37}{5}\right)\sqrt{20}-\dfrac{4}{7}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}+\left(\dfrac{22}{3}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{11}{2}\right)\sqrt{5}\right)+\left(\left(\left(\dfrac{146}{7}\right)\sqrt{5}+\left(-\dfrac{185}{4}\right)\sqrt{5}+\dfrac{1}{4}\right)-\left(\left(\dfrac{74}{5}\right)\sqrt{5}-\dfrac{4}{7}\right)-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}+\dfrac{110}{3}\right)\right)\\
&=&\left(\dfrac{11}{2}\right)\sqrt{5}+\left(\left(\dfrac{146}{7}\right)\sqrt{5}+\left(-\dfrac{185}{4}\right)\sqrt{5}+\dfrac{1}{4}\right)-\left(\left(\dfrac{74}{5}\right)\sqrt{5}-\dfrac{4}{7}\right)-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}+\dfrac{110}{3}\right)\\
&=&\left(\dfrac{443}{140}\right)\sqrt{5}-\dfrac{3011}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{11}{6}\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{73}{7}\right)\sqrt{20}+\left(-\dfrac{37}{4}\right)\sqrt{125}+\dfrac{1}{4}\right)-\left(\left(\dfrac{37}{5}\right)\sqrt{20}-\dfrac{4}{7}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}+\left(\dfrac{22}{3}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{11}{2}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{146}{7}\right)\sqrt{5}+\left(-\dfrac{185}{4}\right)\sqrt{5}+\dfrac{1}{4}\right)-\left(\left(\dfrac{74}{5}\right)\sqrt{5}-\dfrac{4}{7}\right)-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}+\dfrac{110}{3}\right)\right)\\
&=&\left(\left(\dfrac{11}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{327}{140}\right)\sqrt{5}-\dfrac{3011}{84}\right)\\
&=&\left(\dfrac{11}{2}\right)\sqrt{5}+\left(\dfrac{327}{140}\right)\sqrt{5}+\dfrac{3011}{84}\\
&=&\left(\dfrac{1097}{140}\right)\sqrt{5}+\dfrac{3011}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{11}{6}\right)\sqrt{45}\right)\times\left(\left(\left(\dfrac{73}{7}\right)\sqrt{20}+\left(-\dfrac{37}{4}\right)\sqrt{125}+\dfrac{1}{4}\right)-\left(\left(\dfrac{37}{5}\right)\sqrt{20}-\dfrac{4}{7}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}+\left(\dfrac{22}{3}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{11}{2}\right)\sqrt{5}\right)\times\left(\left(\left(\dfrac{146}{7}\right)\sqrt{5}+\left(-\dfrac{185}{4}\right)\sqrt{5}+\dfrac{1}{4}\right)-\left(\left(\dfrac{74}{5}\right)\sqrt{5}-\dfrac{4}{7}\right)-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}+\dfrac{110}{3}\right)\right)\\
&=&\left(\left(\dfrac{11}{2}\right)\sqrt{5}\right)\left(\left(-\dfrac{327}{140}\right)\sqrt{5}-\dfrac{3011}{84}\right)\\
&=&\left(-\dfrac{3597}{280}\right)\sqrt{25}+\left(-\dfrac{33121}{168}\right)\sqrt{5}\\
&=&-\dfrac{3597}{56}+\left(-\dfrac{33121}{168}\right)\sqrt{5}\\
\end{eqnarray*}