L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{17}{9}\right)\sqrt{20}\) et \( Y=\left(\left(\left(4\right)\sqrt{45}\right)-\left(\left(\dfrac{48}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{125}\right)\right)-\left(\left(\left(\dfrac{67}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{31}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{73}{9}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{25}\right)\) . 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{17}{9}\right)\sqrt{20}\right)+\left(\left(\left(\left(4\right)\sqrt{45}\right)-\left(\left(\dfrac{48}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{125}\right)\right)-\left(\left(\left(\dfrac{67}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{31}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{73}{9}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(-\dfrac{34}{9}\right)\sqrt{5}\right)+\left(\left(\left(\left(12\right)\sqrt{5}\right)-\left(\left(48\right)\sqrt{5}\right)-\left(\left(\dfrac{315}{8}\right)\sqrt{5}\right)\right)-\left(67-\left(\left(\dfrac{155}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{73}{3}\right)\sqrt{5}\right)\right)-\dfrac{5}{8}\right)\\
&=&\left(-\dfrac{34}{9}\right)\sqrt{5}+\left(\left(\left(12\right)\sqrt{5}\right)-\left(\left(48\right)\sqrt{5}\right)-\left(\left(\dfrac{315}{8}\right)\sqrt{5}\right)\right)-\left(67-\left(\left(\dfrac{155}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{73}{3}\right)\sqrt{5}\right)\right)-\dfrac{5}{8}\\
&=&\left(-\dfrac{227}{72}\right)\sqrt{5}-\dfrac{541}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{17}{9}\right)\sqrt{20}\right)-\left(\left(\left(\left(4\right)\sqrt{45}\right)-\left(\left(\dfrac{48}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{125}\right)\right)-\left(\left(\left(\dfrac{67}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{31}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{73}{9}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(-\dfrac{34}{9}\right)\sqrt{5}\right)-\left(\left(\left(\left(12\right)\sqrt{5}\right)-\left(\left(48\right)\sqrt{5}\right)-\left(\left(\dfrac{315}{8}\right)\sqrt{5}\right)\right)-\left(67-\left(\left(\dfrac{155}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{73}{3}\right)\sqrt{5}\right)\right)-\dfrac{5}{8}\right)\\
&=&\left(\left(-\dfrac{34}{9}\right)\sqrt{5}\right)-\left(\left(\dfrac{5}{8}\right)\sqrt{5}-\dfrac{541}{8}\right)\\
&=&\left(-\dfrac{34}{9}\right)\sqrt{5}+\left(-\dfrac{5}{8}\right)\sqrt{5}+\dfrac{541}{8}\\
&=&\left(-\dfrac{317}{72}\right)\sqrt{5}+\dfrac{541}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{17}{9}\right)\sqrt{20}\right)\times\left(\left(\left(\left(4\right)\sqrt{45}\right)-\left(\left(\dfrac{48}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{125}\right)\right)-\left(\left(\left(\dfrac{67}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{31}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{73}{9}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(-\dfrac{34}{9}\right)\sqrt{5}\right)\times\left(\left(\left(\left(12\right)\sqrt{5}\right)-\left(\left(48\right)\sqrt{5}\right)-\left(\left(\dfrac{315}{8}\right)\sqrt{5}\right)\right)-\left(67-\left(\left(\dfrac{155}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{73}{3}\right)\sqrt{5}\right)\right)-\dfrac{5}{8}\right)\\
&=&\left(\left(-\dfrac{34}{9}\right)\sqrt{5}\right)\left(\left(\dfrac{5}{8}\right)\sqrt{5}-\dfrac{541}{8}\right)\\
&=&\left(-\dfrac{85}{36}\right)\sqrt{25}+\left(\dfrac{9197}{36}\right)\sqrt{5}\\
&=&-\dfrac{425}{36}+\left(\dfrac{9197}{36}\right)\sqrt{5}\\
\end{eqnarray*}