L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{81}{4}-\left(\left(-\dfrac{35}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{31}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)-\left(\left(0\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{11}{5}\right)\sqrt{4}+\left(\dfrac{56}{3}\right)\sqrt{50}-\dfrac{24}{5}+\left(\dfrac{9}{2}\right)\sqrt{4}\right)\) et \( Y=\left(-\dfrac{1}{7}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{81}{4}-\left(\left(-\dfrac{35}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{31}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)-\left(\left(0\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{11}{5}\right)\sqrt{4}+\left(\dfrac{56}{3}\right)\sqrt{50}-\dfrac{24}{5}+\left(\dfrac{9}{2}\right)\sqrt{4}\right)\right)+\left(\left(-\dfrac{1}{7}\right)\sqrt{50}\right)\\
&=&\left(-\dfrac{81}{4}-\left(\left(-35\right)\sqrt{2}\right)-\left(-\dfrac{62}{7}-\left(\left(27\right)\sqrt{2}\right)-0\right)-\left(\dfrac{22}{5}+\left(\dfrac{280}{3}\right)\sqrt{2}-\dfrac{24}{5}+9\right)\right)+\left(\left(-\dfrac{5}{7}\right)\sqrt{2}\right)\\
&=&-\dfrac{81}{4}-\left(\left(-35\right)\sqrt{2}\right)-\left(-\dfrac{62}{7}-\left(\left(27\right)\sqrt{2}\right)-0\right)-\left(\dfrac{22}{5}+\left(\dfrac{280}{3}\right)\sqrt{2}-\dfrac{24}{5}+9\right)+\left(-\dfrac{5}{7}\right)\sqrt{2}\\
&=&-\dfrac{2799}{140}+\left(-\dfrac{673}{21}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{81}{4}-\left(\left(-\dfrac{35}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{31}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)-\left(\left(0\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{11}{5}\right)\sqrt{4}+\left(\dfrac{56}{3}\right)\sqrt{50}-\dfrac{24}{5}+\left(\dfrac{9}{2}\right)\sqrt{4}\right)\right)-\left(\left(-\dfrac{1}{7}\right)\sqrt{50}\right)\\
&=&\left(-\dfrac{81}{4}-\left(\left(-35\right)\sqrt{2}\right)-\left(-\dfrac{62}{7}-\left(\left(27\right)\sqrt{2}\right)-0\right)-\left(\dfrac{22}{5}+\left(\dfrac{280}{3}\right)\sqrt{2}-\dfrac{24}{5}+9\right)\right)-\left(\left(-\dfrac{5}{7}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{2799}{140}+\left(-\dfrac{94}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{5}{7}\right)\sqrt{2}\right)\\
&=&-\dfrac{2799}{140}+\left(-\dfrac{94}{3}\right)\sqrt{2}+\left(\dfrac{5}{7}\right)\sqrt{2}\\
&=&-\dfrac{2799}{140}+\left(-\dfrac{643}{21}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{81}{4}-\left(\left(-\dfrac{35}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{31}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)-\left(\left(0\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{11}{5}\right)\sqrt{4}+\left(\dfrac{56}{3}\right)\sqrt{50}-\dfrac{24}{5}+\left(\dfrac{9}{2}\right)\sqrt{4}\right)\right)\times\left(\left(-\dfrac{1}{7}\right)\sqrt{50}\right)\\
&=&\left(-\dfrac{81}{4}-\left(\left(-35\right)\sqrt{2}\right)-\left(-\dfrac{62}{7}-\left(\left(27\right)\sqrt{2}\right)-0\right)-\left(\dfrac{22}{5}+\left(\dfrac{280}{3}\right)\sqrt{2}-\dfrac{24}{5}+9\right)\right)\times\left(\left(-\dfrac{5}{7}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{2799}{140}+\left(-\dfrac{94}{3}\right)\sqrt{2}\right)\left(\left(-\dfrac{5}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2799}{196}\right)\sqrt{2}+\left(\dfrac{470}{21}\right)\sqrt{4}\\
&=&\left(\dfrac{2799}{196}\right)\sqrt{2}+\dfrac{940}{21}\\
\end{eqnarray*}