L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{49}{5}\right)\sqrt{12}\) et \( Y=\left(4+\left(-\dfrac{65}{8}\right)\sqrt{75}+\left(-\dfrac{59}{4}\right)\sqrt{9}+3\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{9}\right)\) . 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{49}{5}\right)\sqrt{12}\right)+\left(\left(4+\left(-\dfrac{65}{8}\right)\sqrt{75}+\left(-\dfrac{59}{4}\right)\sqrt{9}+3\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\dfrac{98}{5}\right)\sqrt{3}\right)+\left(\left(4+\left(-\dfrac{325}{8}\right)\sqrt{3}-\dfrac{177}{4}+3\right)+40\right)\\
&=&\left(\dfrac{98}{5}\right)\sqrt{3}+\left(4+\left(-\dfrac{325}{8}\right)\sqrt{3}-\dfrac{177}{4}+3\right)+40\\
&=&\left(-\dfrac{841}{40}\right)\sqrt{3}+\dfrac{11}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{49}{5}\right)\sqrt{12}\right)-\left(\left(4+\left(-\dfrac{65}{8}\right)\sqrt{75}+\left(-\dfrac{59}{4}\right)\sqrt{9}+3\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\dfrac{98}{5}\right)\sqrt{3}\right)-\left(\left(4+\left(-\dfrac{325}{8}\right)\sqrt{3}-\dfrac{177}{4}+3\right)+40\right)\\
&=&\left(\left(\dfrac{98}{5}\right)\sqrt{3}\right)-\left(\dfrac{11}{4}+\left(-\dfrac{325}{8}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{98}{5}\right)\sqrt{3}+-\dfrac{11}{4}+\left(\dfrac{325}{8}\right)\sqrt{3}\\
&=&\left(\dfrac{2409}{40}\right)\sqrt{3}-\dfrac{11}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{49}{5}\right)\sqrt{12}\right)\times\left(\left(4+\left(-\dfrac{65}{8}\right)\sqrt{75}+\left(-\dfrac{59}{4}\right)\sqrt{9}+3\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\dfrac{98}{5}\right)\sqrt{3}\right)\times\left(\left(4+\left(-\dfrac{325}{8}\right)\sqrt{3}-\dfrac{177}{4}+3\right)+40\right)\\
&=&\left(\left(\dfrac{98}{5}\right)\sqrt{3}\right)\left(\dfrac{11}{4}+\left(-\dfrac{325}{8}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{539}{10}\right)\sqrt{3}+\left(-\dfrac{3185}{4}\right)\sqrt{9}\\
&=&\left(\dfrac{539}{10}\right)\sqrt{3}-\dfrac{9555}{4}\\
\end{eqnarray*}