L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(5\right)\sqrt{45}\) et \( Y=\left(\dfrac{20}{9}-\left(\left(-\dfrac{78}{5}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{125}\right)\right)-\left(\left(\left(\dfrac{34}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{71}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{11}{5}\right)\sqrt{25}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(5\right)\sqrt{45}\right)+\left(\left(\dfrac{20}{9}-\left(\left(-\dfrac{78}{5}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{125}\right)\right)-\left(\left(\left(\dfrac{34}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{71}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{11}{5}\right)\sqrt{25}\right)\right)\right)\\
&=&\left(\left(15\right)\sqrt{5}\right)+\left(\left(\dfrac{20}{9}-\left(\left(-\dfrac{156}{5}\right)\sqrt{5}\right)-\left(\left(20\right)\sqrt{5}\right)\right)-\left(\left(\left(34\right)\sqrt{5}\right)-\left(\left(-\dfrac{213}{2}\right)\sqrt{5}\right)-11\right)\right)\\
&=&\left(15\right)\sqrt{5}+\left(\dfrac{20}{9}-\left(\left(-\dfrac{156}{5}\right)\sqrt{5}\right)-\left(\left(20\right)\sqrt{5}\right)\right)-\left(\left(\left(34\right)\sqrt{5}\right)-\left(\left(-\dfrac{213}{2}\right)\sqrt{5}\right)-11\right)\\
&=&\left(-\dfrac{1143}{10}\right)\sqrt{5}+\dfrac{119}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(5\right)\sqrt{45}\right)-\left(\left(\dfrac{20}{9}-\left(\left(-\dfrac{78}{5}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{125}\right)\right)-\left(\left(\left(\dfrac{34}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{71}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{11}{5}\right)\sqrt{25}\right)\right)\right)\\
&=&\left(\left(15\right)\sqrt{5}\right)-\left(\left(\dfrac{20}{9}-\left(\left(-\dfrac{156}{5}\right)\sqrt{5}\right)-\left(\left(20\right)\sqrt{5}\right)\right)-\left(\left(\left(34\right)\sqrt{5}\right)-\left(\left(-\dfrac{213}{2}\right)\sqrt{5}\right)-11\right)\right)\\
&=&\left(\left(15\right)\sqrt{5}\right)-\left(\dfrac{119}{9}+\left(-\dfrac{1293}{10}\right)\sqrt{5}\right)\\
&=&\left(15\right)\sqrt{5}+-\dfrac{119}{9}+\left(\dfrac{1293}{10}\right)\sqrt{5}\\
&=&\left(\dfrac{1443}{10}\right)\sqrt{5}-\dfrac{119}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(5\right)\sqrt{45}\right)\times\left(\left(\dfrac{20}{9}-\left(\left(-\dfrac{78}{5}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{125}\right)\right)-\left(\left(\left(\dfrac{34}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{71}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{11}{5}\right)\sqrt{25}\right)\right)\right)\\
&=&\left(\left(15\right)\sqrt{5}\right)\times\left(\left(\dfrac{20}{9}-\left(\left(-\dfrac{156}{5}\right)\sqrt{5}\right)-\left(\left(20\right)\sqrt{5}\right)\right)-\left(\left(\left(34\right)\sqrt{5}\right)-\left(\left(-\dfrac{213}{2}\right)\sqrt{5}\right)-11\right)\right)\\
&=&\left(\left(15\right)\sqrt{5}\right)\left(\dfrac{119}{9}+\left(-\dfrac{1293}{10}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{595}{3}\right)\sqrt{5}+\left(-\dfrac{3879}{2}\right)\sqrt{25}\\
&=&\left(\dfrac{595}{3}\right)\sqrt{5}-\dfrac{19395}{2}\\
\end{eqnarray*}