L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{1}{2}\right)\sqrt{20}+\left(-7\right)\sqrt{25}+\left(\dfrac{22}{3}\right)\sqrt{25}\right)-\left(\left(\left(9\right)\sqrt{25}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{125}+\left(-\dfrac{34}{5}\right)\sqrt{20}+\left(-\dfrac{15}{2}\right)\sqrt{20}\right)\) et \( Y=\left(-\dfrac{41}{4}\right)\sqrt{20}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{20}+\left(-7\right)\sqrt{25}+\left(\dfrac{22}{3}\right)\sqrt{25}\right)-\left(\left(\left(9\right)\sqrt{25}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{125}+\left(-\dfrac{34}{5}\right)\sqrt{20}+\left(-\dfrac{15}{2}\right)\sqrt{20}\right)\right)+\left(\left(-\dfrac{41}{4}\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-1\right)\sqrt{5}-35+\dfrac{110}{3}\right)-\left(45-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{95}{2}\right)\sqrt{5}+\left(-\dfrac{68}{5}\right)\sqrt{5}+\left(-15\right)\sqrt{5}\right)\right)+\left(\left(-\dfrac{41}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(-1\right)\sqrt{5}-35+\dfrac{110}{3}\right)-\left(45-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{95}{2}\right)\sqrt{5}+\left(-\dfrac{68}{5}\right)\sqrt{5}+\left(-15\right)\sqrt{5}\right)+\left(-\dfrac{41}{2}\right)\sqrt{5}\\
&=&\left(\dfrac{293}{5}\right)\sqrt{5}-\dfrac{130}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{20}+\left(-7\right)\sqrt{25}+\left(\dfrac{22}{3}\right)\sqrt{25}\right)-\left(\left(\left(9\right)\sqrt{25}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{125}+\left(-\dfrac{34}{5}\right)\sqrt{20}+\left(-\dfrac{15}{2}\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-1\right)\sqrt{5}-35+\dfrac{110}{3}\right)-\left(45-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{95}{2}\right)\sqrt{5}+\left(-\dfrac{68}{5}\right)\sqrt{5}+\left(-15\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{41}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{791}{10}\right)\sqrt{5}-\dfrac{130}{3}\right)-\left(\left(-\dfrac{41}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{791}{10}\right)\sqrt{5}-\dfrac{130}{3}+\left(\dfrac{41}{2}\right)\sqrt{5}\\
&=&\left(\dfrac{498}{5}\right)\sqrt{5}-\dfrac{130}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{20}+\left(-7\right)\sqrt{25}+\left(\dfrac{22}{3}\right)\sqrt{25}\right)-\left(\left(\left(9\right)\sqrt{25}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{125}+\left(-\dfrac{34}{5}\right)\sqrt{20}+\left(-\dfrac{15}{2}\right)\sqrt{20}\right)\right)\times\left(\left(-\dfrac{41}{4}\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-1\right)\sqrt{5}-35+\dfrac{110}{3}\right)-\left(45-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{95}{2}\right)\sqrt{5}+\left(-\dfrac{68}{5}\right)\sqrt{5}+\left(-15\right)\sqrt{5}\right)\right)\times\left(\left(-\dfrac{41}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{791}{10}\right)\sqrt{5}-\dfrac{130}{3}\right)\left(\left(-\dfrac{41}{2}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{32431}{20}\right)\sqrt{25}+\left(\dfrac{2665}{3}\right)\sqrt{5}\\
&=&-\dfrac{32431}{4}+\left(\dfrac{2665}{3}\right)\sqrt{5}\\
\end{eqnarray*}