L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-2\right)\sqrt{18}+\left(\dfrac{47}{6}\right)\sqrt{50}-\dfrac{34}{9}\) et \( Y=\left(\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{53}{8}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{8}+4\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-2\right)\sqrt{18}+\left(\dfrac{47}{6}\right)\sqrt{50}-\dfrac{34}{9}\right)+\left(\left(\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{53}{8}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{8}+4\right)\right)\\
&=&\left(\left(-6\right)\sqrt{2}+\left(\dfrac{235}{6}\right)\sqrt{2}-\dfrac{34}{9}\right)+\left(\left(\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{265}{8}\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{2}+4\right)\right)\\
&=&\left(-6\right)\sqrt{2}+\left(\dfrac{235}{6}\right)\sqrt{2}-\dfrac{34}{9}+\left(\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{265}{8}\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{2}+4\right)\\
&=&\left(-\dfrac{45}{8}\right)\sqrt{2}-\dfrac{70}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-2\right)\sqrt{18}+\left(\dfrac{47}{6}\right)\sqrt{50}-\dfrac{34}{9}\right)-\left(\left(\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{53}{8}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{8}+4\right)\right)\\
&=&\left(\left(-6\right)\sqrt{2}+\left(\dfrac{235}{6}\right)\sqrt{2}-\dfrac{34}{9}\right)-\left(\left(\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{265}{8}\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{2}+4\right)\right)\\
&=&\left(\left(\dfrac{199}{6}\right)\sqrt{2}-\dfrac{34}{9}\right)-\left(\left(-\dfrac{931}{24}\right)\sqrt{2}-4\right)\\
&=&\left(\dfrac{199}{6}\right)\sqrt{2}-\dfrac{34}{9}+\left(\dfrac{931}{24}\right)\sqrt{2}+4\\
&=&\left(\dfrac{1727}{24}\right)\sqrt{2}+\dfrac{2}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-2\right)\sqrt{18}+\left(\dfrac{47}{6}\right)\sqrt{50}-\dfrac{34}{9}\right)\times\left(\left(\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{53}{8}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{17}{6}\right)\sqrt{8}+4\right)\right)\\
&=&\left(\left(-6\right)\sqrt{2}+\left(\dfrac{235}{6}\right)\sqrt{2}-\dfrac{34}{9}\right)\times\left(\left(\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{265}{8}\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{2}+4\right)\right)\\
&=&\left(\left(\dfrac{199}{6}\right)\sqrt{2}-\dfrac{34}{9}\right)\left(\left(-\dfrac{931}{24}\right)\sqrt{2}-4\right)\\
&=&\left(-\dfrac{185269}{144}\right)\sqrt{4}+\left(\dfrac{1499}{108}\right)\sqrt{2}+\dfrac{136}{9}\\
&=&-\dfrac{184181}{72}+\left(\dfrac{1499}{108}\right)\sqrt{2}\\
\end{eqnarray*}