L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{43}{5}\right)\sqrt{18}-\dfrac{81}{5}+\left(-\dfrac{32}{3}\right)\sqrt{50}\) et \( Y=\left(\left(-\dfrac{19}{2}\right)\sqrt{18}+\left(-\dfrac{37}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{31}{5}\right)\sqrt{50}+\left(-3\right)\sqrt{4}+\left(-\dfrac{32}{7}\right)\sqrt{50}\right)\) . 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{43}{5}\right)\sqrt{18}-\dfrac{81}{5}+\left(-\dfrac{32}{3}\right)\sqrt{50}\right)+\left(\left(\left(-\dfrac{19}{2}\right)\sqrt{18}+\left(-\dfrac{37}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{31}{5}\right)\sqrt{50}+\left(-3\right)\sqrt{4}+\left(-\dfrac{32}{7}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\dfrac{129}{5}\right)\sqrt{2}-\dfrac{81}{5}+\left(-\dfrac{160}{3}\right)\sqrt{2}\right)+\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{2}+\left(-\dfrac{37}{4}\right)\sqrt{2}\right)-\left(\left(31\right)\sqrt{2}-6+\left(-\dfrac{160}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{129}{5}\right)\sqrt{2}-\dfrac{81}{5}+\left(-\dfrac{160}{3}\right)\sqrt{2}+\left(\left(-\dfrac{57}{2}\right)\sqrt{2}+\left(-\dfrac{37}{4}\right)\sqrt{2}\right)-\left(\left(31\right)\sqrt{2}-6+\left(-\dfrac{160}{7}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{30839}{420}\right)\sqrt{2}-\dfrac{51}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{43}{5}\right)\sqrt{18}-\dfrac{81}{5}+\left(-\dfrac{32}{3}\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{19}{2}\right)\sqrt{18}+\left(-\dfrac{37}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{31}{5}\right)\sqrt{50}+\left(-3\right)\sqrt{4}+\left(-\dfrac{32}{7}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\dfrac{129}{5}\right)\sqrt{2}-\dfrac{81}{5}+\left(-\dfrac{160}{3}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{2}+\left(-\dfrac{37}{4}\right)\sqrt{2}\right)-\left(\left(31\right)\sqrt{2}-6+\left(-\dfrac{160}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{413}{15}\right)\sqrt{2}-\dfrac{81}{5}\right)-\left(\left(-\dfrac{1285}{28}\right)\sqrt{2}+6\right)\\
&=&\left(-\dfrac{413}{15}\right)\sqrt{2}-\dfrac{81}{5}+\left(\dfrac{1285}{28}\right)\sqrt{2}-6\\
&=&\left(\dfrac{7711}{420}\right)\sqrt{2}-\dfrac{111}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{43}{5}\right)\sqrt{18}-\dfrac{81}{5}+\left(-\dfrac{32}{3}\right)\sqrt{50}\right)\times\left(\left(\left(-\dfrac{19}{2}\right)\sqrt{18}+\left(-\dfrac{37}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{31}{5}\right)\sqrt{50}+\left(-3\right)\sqrt{4}+\left(-\dfrac{32}{7}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\dfrac{129}{5}\right)\sqrt{2}-\dfrac{81}{5}+\left(-\dfrac{160}{3}\right)\sqrt{2}\right)\times\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{2}+\left(-\dfrac{37}{4}\right)\sqrt{2}\right)-\left(\left(31\right)\sqrt{2}-6+\left(-\dfrac{160}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{413}{15}\right)\sqrt{2}-\dfrac{81}{5}\right)\left(\left(-\dfrac{1285}{28}\right)\sqrt{2}+6\right)\\
&=&\left(\dfrac{15163}{12}\right)\sqrt{4}+\left(\dfrac{80957}{140}\right)\sqrt{2}-\dfrac{486}{5}\\
&=&\dfrac{72899}{30}+\left(\dfrac{80957}{140}\right)\sqrt{2}\\
\end{eqnarray*}