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 \( X=\left(\dfrac{34}{5}\right)\sqrt{125}+\left(-\dfrac{81}{4}\right)\sqrt{20}+\dfrac{11}{4}-\dfrac{40}{3}\) et \( Y=\left(-\dfrac{13}{3}\right)\sqrt{125}\) . 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{34}{5}\right)\sqrt{125}+\left(-\dfrac{81}{4}\right)\sqrt{20}+\dfrac{11}{4}-\dfrac{40}{3}\right)+\left(\left(-\dfrac{13}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(34\right)\sqrt{5}+\left(-\dfrac{81}{2}\right)\sqrt{5}+\dfrac{11}{4}-\dfrac{40}{3}\right)+\left(\left(-\dfrac{65}{3}\right)\sqrt{5}\right)\\
&=&\left(34\right)\sqrt{5}+\left(-\dfrac{81}{2}\right)\sqrt{5}+\dfrac{11}{4}-\dfrac{40}{3}+\left(-\dfrac{65}{3}\right)\sqrt{5}\\
&=&\left(-\dfrac{169}{6}\right)\sqrt{5}-\dfrac{127}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{34}{5}\right)\sqrt{125}+\left(-\dfrac{81}{4}\right)\sqrt{20}+\dfrac{11}{4}-\dfrac{40}{3}\right)-\left(\left(-\dfrac{13}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(34\right)\sqrt{5}+\left(-\dfrac{81}{2}\right)\sqrt{5}+\dfrac{11}{4}-\dfrac{40}{3}\right)-\left(\left(-\dfrac{65}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{13}{2}\right)\sqrt{5}-\dfrac{127}{12}\right)-\left(\left(-\dfrac{65}{3}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{13}{2}\right)\sqrt{5}-\dfrac{127}{12}+\left(\dfrac{65}{3}\right)\sqrt{5}\\
&=&\left(\dfrac{91}{6}\right)\sqrt{5}-\dfrac{127}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{34}{5}\right)\sqrt{125}+\left(-\dfrac{81}{4}\right)\sqrt{20}+\dfrac{11}{4}-\dfrac{40}{3}\right)\times\left(\left(-\dfrac{13}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(34\right)\sqrt{5}+\left(-\dfrac{81}{2}\right)\sqrt{5}+\dfrac{11}{4}-\dfrac{40}{3}\right)\times\left(\left(-\dfrac{65}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{13}{2}\right)\sqrt{5}-\dfrac{127}{12}\right)\left(\left(-\dfrac{65}{3}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{845}{6}\right)\sqrt{25}+\left(\dfrac{8255}{36}\right)\sqrt{5}\\
&=&\dfrac{4225}{6}+\left(\dfrac{8255}{36}\right)\sqrt{5}\\
\end{eqnarray*}