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 \( X=\left(\dfrac{61}{6}\right)\sqrt{125}\) et \( Y=\left(1\right)\sqrt{45}+\dfrac{65}{4}+\left(-\dfrac{32}{7}\right)\sqrt{45}+\left(-\dfrac{63}{8}\right)\sqrt{45}+\left(-\dfrac{37}{6}\right)\sqrt{45}\) . 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{61}{6}\right)\sqrt{125}\right)+\left(\left(1\right)\sqrt{45}+\dfrac{65}{4}+\left(-\dfrac{32}{7}\right)\sqrt{45}+\left(-\dfrac{63}{8}\right)\sqrt{45}+\left(-\dfrac{37}{6}\right)\sqrt{45}\right)\\
&=&\left(\left(\dfrac{305}{6}\right)\sqrt{5}\right)+\left(\left(3\right)\sqrt{5}+\dfrac{65}{4}+\left(-\dfrac{96}{7}\right)\sqrt{5}+\left(-\dfrac{189}{8}\right)\sqrt{5}+\left(-\dfrac{37}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{305}{6}\right)\sqrt{5}+\left(3\right)\sqrt{5}+\dfrac{65}{4}+\left(-\dfrac{96}{7}\right)\sqrt{5}+\left(-\dfrac{189}{8}\right)\sqrt{5}+\left(-\dfrac{37}{2}\right)\sqrt{5}\\
&=&\left(-\dfrac{337}{168}\right)\sqrt{5}+\dfrac{65}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{61}{6}\right)\sqrt{125}\right)-\left(\left(1\right)\sqrt{45}+\dfrac{65}{4}+\left(-\dfrac{32}{7}\right)\sqrt{45}+\left(-\dfrac{63}{8}\right)\sqrt{45}+\left(-\dfrac{37}{6}\right)\sqrt{45}\right)\\
&=&\left(\left(\dfrac{305}{6}\right)\sqrt{5}\right)-\left(\left(3\right)\sqrt{5}+\dfrac{65}{4}+\left(-\dfrac{96}{7}\right)\sqrt{5}+\left(-\dfrac{189}{8}\right)\sqrt{5}+\left(-\dfrac{37}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{305}{6}\right)\sqrt{5}\right)-\left(\left(-\dfrac{2959}{56}\right)\sqrt{5}+\dfrac{65}{4}\right)\\
&=&\left(\dfrac{305}{6}\right)\sqrt{5}+\left(\dfrac{2959}{56}\right)\sqrt{5}-\dfrac{65}{4}\\
&=&\left(\dfrac{17417}{168}\right)\sqrt{5}-\dfrac{65}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{61}{6}\right)\sqrt{125}\right)\times\left(\left(1\right)\sqrt{45}+\dfrac{65}{4}+\left(-\dfrac{32}{7}\right)\sqrt{45}+\left(-\dfrac{63}{8}\right)\sqrt{45}+\left(-\dfrac{37}{6}\right)\sqrt{45}\right)\\
&=&\left(\left(\dfrac{305}{6}\right)\sqrt{5}\right)\times\left(\left(3\right)\sqrt{5}+\dfrac{65}{4}+\left(-\dfrac{96}{7}\right)\sqrt{5}+\left(-\dfrac{189}{8}\right)\sqrt{5}+\left(-\dfrac{37}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{305}{6}\right)\sqrt{5}\right)\left(\left(-\dfrac{2959}{56}\right)\sqrt{5}+\dfrac{65}{4}\right)\\
&=&\left(-\dfrac{902495}{336}\right)\sqrt{25}+\left(\dfrac{19825}{24}\right)\sqrt{5}\\
&=&-\dfrac{4512475}{336}+\left(\dfrac{19825}{24}\right)\sqrt{5}\\
\end{eqnarray*}