L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{61}{9}\right)\sqrt{125}+\left(-3\right)\sqrt{45}\right)-\left(\left(-\dfrac{4}{7}\right)\sqrt{125}\right)-\dfrac{1}{6}\) et \( Y=\left(\left(4\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{20}\right)+\left(\left(-8\right)\sqrt{45}\right)-\left(\left(\dfrac{61}{7}\right)\sqrt{125}\right)+\dfrac{15}{2}\) . 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{61}{9}\right)\sqrt{125}+\left(-3\right)\sqrt{45}\right)-\left(\left(-\dfrac{4}{7}\right)\sqrt{125}\right)-\dfrac{1}{6}\right)+\left(\left(\left(4\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{20}\right)+\left(\left(-8\right)\sqrt{45}\right)-\left(\left(\dfrac{61}{7}\right)\sqrt{125}\right)+\dfrac{15}{2}\right)\\
&=&\left(\left(\left(\dfrac{305}{9}\right)\sqrt{5}+\left(-9\right)\sqrt{5}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{5}\right)-\dfrac{1}{6}\right)+\left(\left(\left(12\right)\sqrt{5}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)+\left(\left(-24\right)\sqrt{5}\right)-\left(\left(\dfrac{305}{7}\right)\sqrt{5}\right)+\dfrac{15}{2}\right)\\
&=&\left(\left(\dfrac{305}{9}\right)\sqrt{5}+\left(-9\right)\sqrt{5}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{5}\right)-\dfrac{1}{6}+\left(\left(12\right)\sqrt{5}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)+\left(\left(-24\right)\sqrt{5}\right)-\left(\left(\dfrac{305}{7}\right)\sqrt{5}\right)+\dfrac{15}{2}\\
&=&\left(\dfrac{137}{63}\right)\sqrt{5}+\dfrac{22}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{61}{9}\right)\sqrt{125}+\left(-3\right)\sqrt{45}\right)-\left(\left(-\dfrac{4}{7}\right)\sqrt{125}\right)-\dfrac{1}{6}\right)-\left(\left(\left(4\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{20}\right)+\left(\left(-8\right)\sqrt{45}\right)-\left(\left(\dfrac{61}{7}\right)\sqrt{125}\right)+\dfrac{15}{2}\right)\\
&=&\left(\left(\left(\dfrac{305}{9}\right)\sqrt{5}+\left(-9\right)\sqrt{5}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{5}\right)-\dfrac{1}{6}\right)-\left(\left(\left(12\right)\sqrt{5}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)+\left(\left(-24\right)\sqrt{5}\right)-\left(\left(\dfrac{305}{7}\right)\sqrt{5}\right)+\dfrac{15}{2}\right)\\
&=&\left(\left(\dfrac{1748}{63}\right)\sqrt{5}-\dfrac{1}{6}\right)-\left(\left(-\dfrac{179}{7}\right)\sqrt{5}+\dfrac{15}{2}\right)\\
&=&\left(\dfrac{1748}{63}\right)\sqrt{5}-\dfrac{1}{6}+\left(\dfrac{179}{7}\right)\sqrt{5}-\dfrac{15}{2}\\
&=&\left(\dfrac{3359}{63}\right)\sqrt{5}-\dfrac{23}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{61}{9}\right)\sqrt{125}+\left(-3\right)\sqrt{45}\right)-\left(\left(-\dfrac{4}{7}\right)\sqrt{125}\right)-\dfrac{1}{6}\right)\times\left(\left(\left(4\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{20}\right)+\left(\left(-8\right)\sqrt{45}\right)-\left(\left(\dfrac{61}{7}\right)\sqrt{125}\right)+\dfrac{15}{2}\right)\\
&=&\left(\left(\left(\dfrac{305}{9}\right)\sqrt{5}+\left(-9\right)\sqrt{5}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{5}\right)-\dfrac{1}{6}\right)\times\left(\left(\left(12\right)\sqrt{5}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{5}\right)+\left(\left(-24\right)\sqrt{5}\right)-\left(\left(\dfrac{305}{7}\right)\sqrt{5}\right)+\dfrac{15}{2}\right)\\
&=&\left(\left(\dfrac{1748}{63}\right)\sqrt{5}-\dfrac{1}{6}\right)\left(\left(-\dfrac{179}{7}\right)\sqrt{5}+\dfrac{15}{2}\right)\\
&=&\left(-\dfrac{312892}{441}\right)\sqrt{25}+\left(\dfrac{2973}{14}\right)\sqrt{5}-\dfrac{5}{4}\\
&=&-\dfrac{6260045}{1764}+\left(\dfrac{2973}{14}\right)\sqrt{5}\\
\end{eqnarray*}