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 \( X=\left(\dfrac{69}{8}\right)\sqrt{45}+\left(-\dfrac{1}{8}\right)\sqrt{25}+\left(-8\right)\sqrt{125}+4-\left(\left(-9\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{125}\right)\) et \( Y=\left(\left(-\dfrac{65}{6}\right)\sqrt{25}\right)-\left(\dfrac{72}{7}-\left(\left(-8\right)\sqrt{25}\right)\right)-\left(\left(1\right)\sqrt{25}\right)\) . 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{69}{8}\right)\sqrt{45}+\left(-\dfrac{1}{8}\right)\sqrt{25}+\left(-8\right)\sqrt{125}+4-\left(\left(-9\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{125}\right)\right)+\left(\left(\left(-\dfrac{65}{6}\right)\sqrt{25}\right)-\left(\dfrac{72}{7}-\left(\left(-8\right)\sqrt{25}\right)\right)-\left(\left(1\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{207}{8}\right)\sqrt{5}-\dfrac{5}{8}+\left(-40\right)\sqrt{5}+4-\left(\left(-27\right)\sqrt{5}\right)-\left(\left(-\dfrac{85}{3}\right)\sqrt{5}\right)\right)+\left(-\dfrac{325}{6}-\left(\dfrac{72}{7}+40\right)-5\right)\\
&=&\left(\dfrac{207}{8}\right)\sqrt{5}-\dfrac{5}{8}+\left(-40\right)\sqrt{5}+4-\left(\left(-27\right)\sqrt{5}\right)-\left(\left(-\dfrac{85}{3}\right)\sqrt{5}\right)-\dfrac{325}{6}-\left(\dfrac{72}{7}+40\right)-5\\
&=&\left(\dfrac{989}{24}\right)\sqrt{5}-\dfrac{17821}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{69}{8}\right)\sqrt{45}+\left(-\dfrac{1}{8}\right)\sqrt{25}+\left(-8\right)\sqrt{125}+4-\left(\left(-9\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{65}{6}\right)\sqrt{25}\right)-\left(\dfrac{72}{7}-\left(\left(-8\right)\sqrt{25}\right)\right)-\left(\left(1\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{207}{8}\right)\sqrt{5}-\dfrac{5}{8}+\left(-40\right)\sqrt{5}+4-\left(\left(-27\right)\sqrt{5}\right)-\left(\left(-\dfrac{85}{3}\right)\sqrt{5}\right)\right)-\left(-\dfrac{325}{6}-\left(\dfrac{72}{7}+40\right)-5\right)\\
&=&\left(\left(\dfrac{989}{24}\right)\sqrt{5}+\dfrac{27}{8}\right)-\left(-\dfrac{4597}{42}\right)\\
&=&\left(\dfrac{989}{24}\right)\sqrt{5}+\dfrac{27}{8}+\dfrac{4597}{42}\\
&=&\left(\dfrac{989}{24}\right)\sqrt{5}+\dfrac{18955}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{69}{8}\right)\sqrt{45}+\left(-\dfrac{1}{8}\right)\sqrt{25}+\left(-8\right)\sqrt{125}+4-\left(\left(-9\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{125}\right)\right)\times\left(\left(\left(-\dfrac{65}{6}\right)\sqrt{25}\right)-\left(\dfrac{72}{7}-\left(\left(-8\right)\sqrt{25}\right)\right)-\left(\left(1\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{207}{8}\right)\sqrt{5}-\dfrac{5}{8}+\left(-40\right)\sqrt{5}+4-\left(\left(-27\right)\sqrt{5}\right)-\left(\left(-\dfrac{85}{3}\right)\sqrt{5}\right)\right)\times\left(-\dfrac{325}{6}-\left(\dfrac{72}{7}+40\right)-5\right)\\
&=&\left(\left(\dfrac{989}{24}\right)\sqrt{5}+\dfrac{27}{8}\right)\left(-\dfrac{4597}{42}\right)\\
&=&\left(-\dfrac{4546433}{1008}\right)\sqrt{5}-\dfrac{41373}{112}\\
&=&\left(-\dfrac{4546433}{1008}\right)\sqrt{5}-\dfrac{41373}{112}\\
\end{eqnarray*}