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 \( X=\dfrac{5}{4}\) et \( Y=\left(\dfrac{25}{4}\right)\sqrt{125}+1+\dfrac{35}{4}+\left(\dfrac{2}{9}\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{25}+\left(4\right)\sqrt{25}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{5}{4}\right)+\left(\left(\dfrac{25}{4}\right)\sqrt{125}+1+\dfrac{35}{4}+\left(\dfrac{2}{9}\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{25}+\left(4\right)\sqrt{25}\right)\\
&=&\left(\dfrac{5}{4}\right)+\left(\left(\dfrac{125}{4}\right)\sqrt{5}+1+\dfrac{35}{4}+\dfrac{10}{9}+\dfrac{115}{3}+20\right)\\
&=&\dfrac{5}{4}+\left(\dfrac{125}{4}\right)\sqrt{5}+1+\dfrac{35}{4}+\dfrac{10}{9}+\dfrac{115}{3}+20\\
&=&\dfrac{634}{9}+\left(\dfrac{125}{4}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{5}{4}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{125}+1+\dfrac{35}{4}+\left(\dfrac{2}{9}\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{25}+\left(4\right)\sqrt{25}\right)\\
&=&\left(\dfrac{5}{4}\right)-\left(\left(\dfrac{125}{4}\right)\sqrt{5}+1+\dfrac{35}{4}+\dfrac{10}{9}+\dfrac{115}{3}+20\right)\\
&=&\left(\dfrac{5}{4}\right)-\left(\left(\dfrac{125}{4}\right)\sqrt{5}+\dfrac{2491}{36}\right)\\
&=&\dfrac{5}{4}+\left(-\dfrac{125}{4}\right)\sqrt{5}-\dfrac{2491}{36}\\
&=&-\dfrac{1223}{18}+\left(-\dfrac{125}{4}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{5}{4}\right)\times\left(\left(\dfrac{25}{4}\right)\sqrt{125}+1+\dfrac{35}{4}+\left(\dfrac{2}{9}\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{25}+\left(4\right)\sqrt{25}\right)\\
&=&\left(\dfrac{5}{4}\right)\times\left(\left(\dfrac{125}{4}\right)\sqrt{5}+1+\dfrac{35}{4}+\dfrac{10}{9}+\dfrac{115}{3}+20\right)\\
&=&\left(\dfrac{5}{4}\right)\left(\left(\dfrac{125}{4}\right)\sqrt{5}+\dfrac{2491}{36}\right)\\
&=&\left(\dfrac{625}{16}\right)\sqrt{5}+\dfrac{12455}{144}\\
&=&\left(\dfrac{625}{16}\right)\sqrt{5}+\dfrac{12455}{144}\\
\end{eqnarray*}