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 \( X=\left(\dfrac{31}{3}\right)\sqrt{125}\) et \( Y=\left(\left(-6\right)\sqrt{25}+\left(-\dfrac{79}{8}\right)\sqrt{125}+\left(6\right)\sqrt{125}\right)-\left(\left(-\dfrac{47}{9}\right)\sqrt{25}\right)\) . 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{31}{3}\right)\sqrt{125}\right)+\left(\left(\left(-6\right)\sqrt{25}+\left(-\dfrac{79}{8}\right)\sqrt{125}+\left(6\right)\sqrt{125}\right)-\left(\left(-\dfrac{47}{9}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{155}{3}\right)\sqrt{5}\right)+\left(\left(-30+\left(-\dfrac{395}{8}\right)\sqrt{5}+\left(30\right)\sqrt{5}\right)+\dfrac{235}{9}\right)\\
&=&\left(\dfrac{155}{3}\right)\sqrt{5}+\left(-30+\left(-\dfrac{395}{8}\right)\sqrt{5}+\left(30\right)\sqrt{5}\right)+\dfrac{235}{9}\\
&=&\left(\dfrac{775}{24}\right)\sqrt{5}-\dfrac{35}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{31}{3}\right)\sqrt{125}\right)-\left(\left(\left(-6\right)\sqrt{25}+\left(-\dfrac{79}{8}\right)\sqrt{125}+\left(6\right)\sqrt{125}\right)-\left(\left(-\dfrac{47}{9}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{155}{3}\right)\sqrt{5}\right)-\left(\left(-30+\left(-\dfrac{395}{8}\right)\sqrt{5}+\left(30\right)\sqrt{5}\right)+\dfrac{235}{9}\right)\\
&=&\left(\left(\dfrac{155}{3}\right)\sqrt{5}\right)-\left(-\dfrac{35}{9}+\left(-\dfrac{155}{8}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{155}{3}\right)\sqrt{5}+\dfrac{35}{9}+\left(\dfrac{155}{8}\right)\sqrt{5}\\
&=&\left(\dfrac{1705}{24}\right)\sqrt{5}+\dfrac{35}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{31}{3}\right)\sqrt{125}\right)\times\left(\left(\left(-6\right)\sqrt{25}+\left(-\dfrac{79}{8}\right)\sqrt{125}+\left(6\right)\sqrt{125}\right)-\left(\left(-\dfrac{47}{9}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{155}{3}\right)\sqrt{5}\right)\times\left(\left(-30+\left(-\dfrac{395}{8}\right)\sqrt{5}+\left(30\right)\sqrt{5}\right)+\dfrac{235}{9}\right)\\
&=&\left(\left(\dfrac{155}{3}\right)\sqrt{5}\right)\left(-\dfrac{35}{9}+\left(-\dfrac{155}{8}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{5425}{27}\right)\sqrt{5}+\left(-\dfrac{24025}{24}\right)\sqrt{25}\\
&=&\left(-\dfrac{5425}{27}\right)\sqrt{5}-\dfrac{120125}{24}\\
\end{eqnarray*}