L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{77}{3}+\left(-\dfrac{75}{4}\right)\sqrt{8}+\left(3\right)\sqrt{4}\) et \( Y=\left(\dfrac{29}{4}\right)\sqrt{50}+\left(\dfrac{37}{8}\right)\sqrt{18}-\dfrac{15}{8}-7-\left(\left(-5\right)\sqrt{8}\right)-\left(\left(-\dfrac{79}{5}\right)\sqrt{8}\right)+\left(\dfrac{76}{7}\right)\sqrt{4}+8+\left(-\dfrac{2}{3}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{77}{3}+\left(-\dfrac{75}{4}\right)\sqrt{8}+\left(3\right)\sqrt{4}\right)+\left(\left(\dfrac{29}{4}\right)\sqrt{50}+\left(\dfrac{37}{8}\right)\sqrt{18}-\dfrac{15}{8}-7-\left(\left(-5\right)\sqrt{8}\right)-\left(\left(-\dfrac{79}{5}\right)\sqrt{8}\right)+\left(\dfrac{76}{7}\right)\sqrt{4}+8+\left(-\dfrac{2}{3}\right)\sqrt{18}\right)\\
&=&\left(-\dfrac{77}{3}+\left(-\dfrac{75}{2}\right)\sqrt{2}+6\right)+\left(\left(\dfrac{145}{4}\right)\sqrt{2}+\left(\dfrac{111}{8}\right)\sqrt{2}-\dfrac{15}{8}-7-\left(\left(-10\right)\sqrt{2}\right)-\left(\left(-\dfrac{158}{5}\right)\sqrt{2}\right)+\dfrac{152}{7}+8+\left(-2\right)\sqrt{2}\right)\\
&=&-\dfrac{77}{3}+\left(-\dfrac{75}{2}\right)\sqrt{2}+6+\left(\dfrac{145}{4}\right)\sqrt{2}+\left(\dfrac{111}{8}\right)\sqrt{2}-\dfrac{15}{8}-7-\left(\left(-10\right)\sqrt{2}\right)-\left(\left(-\dfrac{158}{5}\right)\sqrt{2}\right)+\dfrac{152}{7}+8+\left(-2\right)\sqrt{2}\\
&=&\dfrac{197}{168}+\left(\dfrac{2089}{40}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{77}{3}+\left(-\dfrac{75}{4}\right)\sqrt{8}+\left(3\right)\sqrt{4}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{50}+\left(\dfrac{37}{8}\right)\sqrt{18}-\dfrac{15}{8}-7-\left(\left(-5\right)\sqrt{8}\right)-\left(\left(-\dfrac{79}{5}\right)\sqrt{8}\right)+\left(\dfrac{76}{7}\right)\sqrt{4}+8+\left(-\dfrac{2}{3}\right)\sqrt{18}\right)\\
&=&\left(-\dfrac{77}{3}+\left(-\dfrac{75}{2}\right)\sqrt{2}+6\right)-\left(\left(\dfrac{145}{4}\right)\sqrt{2}+\left(\dfrac{111}{8}\right)\sqrt{2}-\dfrac{15}{8}-7-\left(\left(-10\right)\sqrt{2}\right)-\left(\left(-\dfrac{158}{5}\right)\sqrt{2}\right)+\dfrac{152}{7}+8+\left(-2\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{59}{3}+\left(-\dfrac{75}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{3589}{40}\right)\sqrt{2}+\dfrac{1167}{56}\right)\\
&=&-\dfrac{59}{3}+\left(-\dfrac{75}{2}\right)\sqrt{2}+\left(-\dfrac{3589}{40}\right)\sqrt{2}-\dfrac{1167}{56}\\
&=&-\dfrac{6805}{168}+\left(-\dfrac{5089}{40}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{77}{3}+\left(-\dfrac{75}{4}\right)\sqrt{8}+\left(3\right)\sqrt{4}\right)\times\left(\left(\dfrac{29}{4}\right)\sqrt{50}+\left(\dfrac{37}{8}\right)\sqrt{18}-\dfrac{15}{8}-7-\left(\left(-5\right)\sqrt{8}\right)-\left(\left(-\dfrac{79}{5}\right)\sqrt{8}\right)+\left(\dfrac{76}{7}\right)\sqrt{4}+8+\left(-\dfrac{2}{3}\right)\sqrt{18}\right)\\
&=&\left(-\dfrac{77}{3}+\left(-\dfrac{75}{2}\right)\sqrt{2}+6\right)\times\left(\left(\dfrac{145}{4}\right)\sqrt{2}+\left(\dfrac{111}{8}\right)\sqrt{2}-\dfrac{15}{8}-7-\left(\left(-10\right)\sqrt{2}\right)-\left(\left(-\dfrac{158}{5}\right)\sqrt{2}\right)+\dfrac{152}{7}+8+\left(-2\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{59}{3}+\left(-\dfrac{75}{2}\right)\sqrt{2}\right)\left(\left(\dfrac{3589}{40}\right)\sqrt{2}+\dfrac{1167}{56}\right)\\
&=&\left(-\dfrac{4277389}{1680}\right)\sqrt{2}-\dfrac{22951}{56}+\left(-\dfrac{53835}{16}\right)\sqrt{4}\\
&=&\left(-\dfrac{4277389}{1680}\right)\sqrt{2}-\dfrac{99949}{14}\\
\end{eqnarray*}