L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{53}{9}\right)\sqrt{50}\) et \( Y=-\dfrac{31}{7}+\left(\dfrac{20}{9}\right)\sqrt{4}+\left(-8\right)\sqrt{18}+\left(-\dfrac{25}{9}\right)\sqrt{18}-\dfrac{31}{7}+\left(0\right)\sqrt{8}+\left(-\dfrac{17}{3}\right)\sqrt{4}+\dfrac{51}{5}+\left(-\dfrac{25}{3}\right)\sqrt{18}+\dfrac{37}{3}+\left(-4\right)\sqrt{50}+\left(\dfrac{19}{3}\right)\sqrt{50}\) . 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{53}{9}\right)\sqrt{50}\right)+\left(-\dfrac{31}{7}+\left(\dfrac{20}{9}\right)\sqrt{4}+\left(-8\right)\sqrt{18}+\left(-\dfrac{25}{9}\right)\sqrt{18}-\dfrac{31}{7}+\left(0\right)\sqrt{8}+\left(-\dfrac{17}{3}\right)\sqrt{4}+\dfrac{51}{5}+\left(-\dfrac{25}{3}\right)\sqrt{18}+\dfrac{37}{3}+\left(-4\right)\sqrt{50}+\left(\dfrac{19}{3}\right)\sqrt{50}\right)\\
&=&\left(\left(-\dfrac{265}{9}\right)\sqrt{2}\right)+\left(-\dfrac{31}{7}+\dfrac{40}{9}+\left(-24\right)\sqrt{2}+\left(-\dfrac{25}{3}\right)\sqrt{2}-\dfrac{31}{7}+\left(0\right)\sqrt{2}-\dfrac{34}{3}+\dfrac{51}{5}+\left(-25\right)\sqrt{2}+\dfrac{37}{3}+\left(-20\right)\sqrt{2}+\left(\dfrac{95}{3}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{265}{9}\right)\sqrt{2}-\dfrac{31}{7}+\dfrac{40}{9}+\left(-24\right)\sqrt{2}+\left(-\dfrac{25}{3}\right)\sqrt{2}-\dfrac{31}{7}+\left(0\right)\sqrt{2}-\dfrac{34}{3}+\dfrac{51}{5}+\left(-25\right)\sqrt{2}+\dfrac{37}{3}+\left(-20\right)\sqrt{2}+\left(\dfrac{95}{3}\right)\sqrt{2}\\
&=&\left(-\dfrac{676}{9}\right)\sqrt{2}+\dfrac{2138}{315}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{53}{9}\right)\sqrt{50}\right)-\left(-\dfrac{31}{7}+\left(\dfrac{20}{9}\right)\sqrt{4}+\left(-8\right)\sqrt{18}+\left(-\dfrac{25}{9}\right)\sqrt{18}-\dfrac{31}{7}+\left(0\right)\sqrt{8}+\left(-\dfrac{17}{3}\right)\sqrt{4}+\dfrac{51}{5}+\left(-\dfrac{25}{3}\right)\sqrt{18}+\dfrac{37}{3}+\left(-4\right)\sqrt{50}+\left(\dfrac{19}{3}\right)\sqrt{50}\right)\\
&=&\left(\left(-\dfrac{265}{9}\right)\sqrt{2}\right)-\left(-\dfrac{31}{7}+\dfrac{40}{9}+\left(-24\right)\sqrt{2}+\left(-\dfrac{25}{3}\right)\sqrt{2}-\dfrac{31}{7}+\left(0\right)\sqrt{2}-\dfrac{34}{3}+\dfrac{51}{5}+\left(-25\right)\sqrt{2}+\dfrac{37}{3}+\left(-20\right)\sqrt{2}+\left(\dfrac{95}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{265}{9}\right)\sqrt{2}\right)-\left(\dfrac{2138}{315}+\left(-\dfrac{137}{3}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{265}{9}\right)\sqrt{2}+-\dfrac{2138}{315}+\left(\dfrac{137}{3}\right)\sqrt{2}\\
&=&\left(\dfrac{146}{9}\right)\sqrt{2}-\dfrac{2138}{315}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{53}{9}\right)\sqrt{50}\right)\times\left(-\dfrac{31}{7}+\left(\dfrac{20}{9}\right)\sqrt{4}+\left(-8\right)\sqrt{18}+\left(-\dfrac{25}{9}\right)\sqrt{18}-\dfrac{31}{7}+\left(0\right)\sqrt{8}+\left(-\dfrac{17}{3}\right)\sqrt{4}+\dfrac{51}{5}+\left(-\dfrac{25}{3}\right)\sqrt{18}+\dfrac{37}{3}+\left(-4\right)\sqrt{50}+\left(\dfrac{19}{3}\right)\sqrt{50}\right)\\
&=&\left(\left(-\dfrac{265}{9}\right)\sqrt{2}\right)\times\left(-\dfrac{31}{7}+\dfrac{40}{9}+\left(-24\right)\sqrt{2}+\left(-\dfrac{25}{3}\right)\sqrt{2}-\dfrac{31}{7}+\left(0\right)\sqrt{2}-\dfrac{34}{3}+\dfrac{51}{5}+\left(-25\right)\sqrt{2}+\dfrac{37}{3}+\left(-20\right)\sqrt{2}+\left(\dfrac{95}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{265}{9}\right)\sqrt{2}\right)\left(\dfrac{2138}{315}+\left(-\dfrac{137}{3}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{113314}{567}\right)\sqrt{2}+\left(\dfrac{36305}{27}\right)\sqrt{4}\\
&=&\left(-\dfrac{113314}{567}\right)\sqrt{2}+\dfrac{72610}{27}\\
\end{eqnarray*}