L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{4}{9}\right)\sqrt{45}\) et \( Y=\left(\left(\left(7\right)\sqrt{20}\right)-\left(\left(-1\right)\sqrt{20}\right)-\left(\left(-\dfrac{74}{5}\right)\sqrt{125}\right)\right)-\left(-\dfrac{9}{7}-\left(\left(4\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{125}\right)\) . 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{4}{9}\right)\sqrt{45}\right)+\left(\left(\left(\left(7\right)\sqrt{20}\right)-\left(\left(-1\right)\sqrt{20}\right)-\left(\left(-\dfrac{74}{5}\right)\sqrt{125}\right)\right)-\left(-\dfrac{9}{7}-\left(\left(4\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(-\dfrac{4}{3}\right)\sqrt{5}\right)+\left(\left(\left(\left(14\right)\sqrt{5}\right)-\left(\left(-2\right)\sqrt{5}\right)-\left(\left(-74\right)\sqrt{5}\right)\right)-\left(-\dfrac{9}{7}-\left(\left(12\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{215}{9}\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{4}{3}\right)\sqrt{5}+\left(\left(\left(14\right)\sqrt{5}\right)-\left(\left(-2\right)\sqrt{5}\right)-\left(\left(-74\right)\sqrt{5}\right)\right)-\left(-\dfrac{9}{7}-\left(\left(12\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{215}{9}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{691}{9}\right)\sqrt{5}+\dfrac{9}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{4}{9}\right)\sqrt{45}\right)-\left(\left(\left(\left(7\right)\sqrt{20}\right)-\left(\left(-1\right)\sqrt{20}\right)-\left(\left(-\dfrac{74}{5}\right)\sqrt{125}\right)\right)-\left(-\dfrac{9}{7}-\left(\left(4\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(-\dfrac{4}{3}\right)\sqrt{5}\right)-\left(\left(\left(\left(14\right)\sqrt{5}\right)-\left(\left(-2\right)\sqrt{5}\right)-\left(\left(-74\right)\sqrt{5}\right)\right)-\left(-\dfrac{9}{7}-\left(\left(12\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{215}{9}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{4}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{703}{9}\right)\sqrt{5}+\dfrac{9}{7}\right)\\
&=&\left(-\dfrac{4}{3}\right)\sqrt{5}+\left(-\dfrac{703}{9}\right)\sqrt{5}-\dfrac{9}{7}\\
&=&\left(-\dfrac{715}{9}\right)\sqrt{5}-\dfrac{9}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{4}{9}\right)\sqrt{45}\right)\times\left(\left(\left(\left(7\right)\sqrt{20}\right)-\left(\left(-1\right)\sqrt{20}\right)-\left(\left(-\dfrac{74}{5}\right)\sqrt{125}\right)\right)-\left(-\dfrac{9}{7}-\left(\left(4\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(-\dfrac{4}{3}\right)\sqrt{5}\right)\times\left(\left(\left(\left(14\right)\sqrt{5}\right)-\left(\left(-2\right)\sqrt{5}\right)-\left(\left(-74\right)\sqrt{5}\right)\right)-\left(-\dfrac{9}{7}-\left(\left(12\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{215}{9}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{4}{3}\right)\sqrt{5}\right)\left(\left(\dfrac{703}{9}\right)\sqrt{5}+\dfrac{9}{7}\right)\\
&=&\left(-\dfrac{2812}{27}\right)\sqrt{25}+\left(-\dfrac{12}{7}\right)\sqrt{5}\\
&=&-\dfrac{14060}{27}+\left(-\dfrac{12}{7}\right)\sqrt{5}\\
\end{eqnarray*}