L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{23}{3}\right)\sqrt{18}\) et \( Y=\left(\left(-3\right)\sqrt{8}+\dfrac{55}{3}\right)-\left(\left(-8\right)\sqrt{4}+\left(-\dfrac{66}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{29}{8}\right)\sqrt{8}\right)-\dfrac{1}{5}-\left(\left(\dfrac{53}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{37}{8}\right)\sqrt{4}\right)\right)-\left(\left(-2\right)\sqrt{18}+\left(7\right)\sqrt{50}+\dfrac{1}{5}+\left(-\dfrac{66}{7}\right)\sqrt{18}\right)\) . 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{23}{3}\right)\sqrt{18}\right)+\left(\left(\left(-3\right)\sqrt{8}+\dfrac{55}{3}\right)-\left(\left(-8\right)\sqrt{4}+\left(-\dfrac{66}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{29}{8}\right)\sqrt{8}\right)-\dfrac{1}{5}-\left(\left(\dfrac{53}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{37}{8}\right)\sqrt{4}\right)\right)-\left(\left(-2\right)\sqrt{18}+\left(7\right)\sqrt{50}+\dfrac{1}{5}+\left(-\dfrac{66}{7}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(23\right)\sqrt{2}\right)+\left(\left(\left(-6\right)\sqrt{2}+\dfrac{55}{3}\right)-\left(-16+\left(-\dfrac{330}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{29}{4}\right)\sqrt{2}\right)-\dfrac{1}{5}-\dfrac{106}{3}+\dfrac{37}{4}\right)-\left(\left(-6\right)\sqrt{2}+\left(35\right)\sqrt{2}+\dfrac{1}{5}+\left(-\dfrac{198}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(23\right)\sqrt{2}+\left(\left(-6\right)\sqrt{2}+\dfrac{55}{3}\right)-\left(-16+\left(-\dfrac{330}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{29}{4}\right)\sqrt{2}\right)-\dfrac{1}{5}-\dfrac{106}{3}+\dfrac{37}{4}\right)-\left(\left(-6\right)\sqrt{2}+\left(35\right)\sqrt{2}+\dfrac{1}{5}+\left(-\dfrac{198}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{999}{28}\right)\sqrt{2}+\dfrac{725}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{23}{3}\right)\sqrt{18}\right)-\left(\left(\left(-3\right)\sqrt{8}+\dfrac{55}{3}\right)-\left(\left(-8\right)\sqrt{4}+\left(-\dfrac{66}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{29}{8}\right)\sqrt{8}\right)-\dfrac{1}{5}-\left(\left(\dfrac{53}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{37}{8}\right)\sqrt{4}\right)\right)-\left(\left(-2\right)\sqrt{18}+\left(7\right)\sqrt{50}+\dfrac{1}{5}+\left(-\dfrac{66}{7}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(23\right)\sqrt{2}\right)-\left(\left(\left(-6\right)\sqrt{2}+\dfrac{55}{3}\right)-\left(-16+\left(-\dfrac{330}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{29}{4}\right)\sqrt{2}\right)-\dfrac{1}{5}-\dfrac{106}{3}+\dfrac{37}{4}\right)-\left(\left(-6\right)\sqrt{2}+\left(35\right)\sqrt{2}+\dfrac{1}{5}+\left(-\dfrac{198}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(23\right)\sqrt{2}\right)-\left(\left(\dfrac{355}{28}\right)\sqrt{2}+\dfrac{725}{12}\right)\\
&=&\left(23\right)\sqrt{2}+\left(-\dfrac{355}{28}\right)\sqrt{2}-\dfrac{725}{12}\\
&=&\left(\dfrac{289}{28}\right)\sqrt{2}-\dfrac{725}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{23}{3}\right)\sqrt{18}\right)\times\left(\left(\left(-3\right)\sqrt{8}+\dfrac{55}{3}\right)-\left(\left(-8\right)\sqrt{4}+\left(-\dfrac{66}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{29}{8}\right)\sqrt{8}\right)-\dfrac{1}{5}-\left(\left(\dfrac{53}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{37}{8}\right)\sqrt{4}\right)\right)-\left(\left(-2\right)\sqrt{18}+\left(7\right)\sqrt{50}+\dfrac{1}{5}+\left(-\dfrac{66}{7}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(23\right)\sqrt{2}\right)\times\left(\left(\left(-6\right)\sqrt{2}+\dfrac{55}{3}\right)-\left(-16+\left(-\dfrac{330}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{29}{4}\right)\sqrt{2}\right)-\dfrac{1}{5}-\dfrac{106}{3}+\dfrac{37}{4}\right)-\left(\left(-6\right)\sqrt{2}+\left(35\right)\sqrt{2}+\dfrac{1}{5}+\left(-\dfrac{198}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(23\right)\sqrt{2}\right)\left(\left(\dfrac{355}{28}\right)\sqrt{2}+\dfrac{725}{12}\right)\\
&=&\left(\dfrac{8165}{28}\right)\sqrt{4}+\left(\dfrac{16675}{12}\right)\sqrt{2}\\
&=&\dfrac{8165}{14}+\left(\dfrac{16675}{12}\right)\sqrt{2}\\
\end{eqnarray*}