L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{17}{8}\right)\sqrt{18}\right)-\left(\left(-2\right)\sqrt{50}\right)-\left(\left(\dfrac{33}{5}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{4}+\left(-\dfrac{23}{2}\right)\sqrt{4}+\left(\dfrac{77}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{18}\right)-\left(\left(\left(5\right)\sqrt{50}\right)+\dfrac{25}{4}\right)\) et \( Y=\left(-7\right)\sqrt{8}+\left(\left(\dfrac{21}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{4}\right)+\left(-\dfrac{21}{2}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(\dfrac{17}{8}\right)\sqrt{18}\right)-\left(\left(-2\right)\sqrt{50}\right)-\left(\left(\dfrac{33}{5}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{4}+\left(-\dfrac{23}{2}\right)\sqrt{4}+\left(\dfrac{77}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{18}\right)-\left(\left(\left(5\right)\sqrt{50}\right)+\dfrac{25}{4}\right)\right)+\left(\left(-7\right)\sqrt{8}+\left(\left(\dfrac{21}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{4}\right)+\left(-\dfrac{21}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\left(\dfrac{51}{8}\right)\sqrt{2}\right)-\left(\left(-10\right)\sqrt{2}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{2}\right)\right)-\left(-\dfrac{50}{3}-23+\left(\dfrac{385}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{123}{7}\right)\sqrt{2}\right)-\left(\left(\left(25\right)\sqrt{2}\right)+\dfrac{25}{4}\right)\right)+\left(\left(-14\right)\sqrt{2}+\left(\left(\dfrac{105}{2}\right)\sqrt{2}\right)-19-\dfrac{11}{2}+\left(-\dfrac{105}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(\left(\dfrac{51}{8}\right)\sqrt{2}\right)-\left(\left(-10\right)\sqrt{2}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{2}\right)\right)-\left(-\dfrac{50}{3}-23+\left(\dfrac{385}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{123}{7}\right)\sqrt{2}\right)-\left(\left(\left(25\right)\sqrt{2}\right)+\dfrac{25}{4}\right)+\left(-14\right)\sqrt{2}+\left(\left(\dfrac{105}{2}\right)\sqrt{2}\right)-19-\dfrac{11}{2}+\left(-\dfrac{105}{2}\right)\sqrt{2}\\
&=&\left(-\dfrac{9293}{140}\right)\sqrt{2}+\dfrac{107}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{17}{8}\right)\sqrt{18}\right)-\left(\left(-2\right)\sqrt{50}\right)-\left(\left(\dfrac{33}{5}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{4}+\left(-\dfrac{23}{2}\right)\sqrt{4}+\left(\dfrac{77}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{18}\right)-\left(\left(\left(5\right)\sqrt{50}\right)+\dfrac{25}{4}\right)\right)-\left(\left(-7\right)\sqrt{8}+\left(\left(\dfrac{21}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{4}\right)+\left(-\dfrac{21}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\left(\dfrac{51}{8}\right)\sqrt{2}\right)-\left(\left(-10\right)\sqrt{2}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{2}\right)\right)-\left(-\dfrac{50}{3}-23+\left(\dfrac{385}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{123}{7}\right)\sqrt{2}\right)-\left(\left(\left(25\right)\sqrt{2}\right)+\dfrac{25}{4}\right)\right)-\left(\left(-14\right)\sqrt{2}+\left(\left(\dfrac{105}{2}\right)\sqrt{2}\right)-19-\dfrac{11}{2}+\left(-\dfrac{105}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{7333}{140}\right)\sqrt{2}+\dfrac{401}{12}\right)-\left(\left(-14\right)\sqrt{2}-\dfrac{49}{2}\right)\\
&=&\left(-\dfrac{7333}{140}\right)\sqrt{2}+\dfrac{401}{12}+\left(14\right)\sqrt{2}+\dfrac{49}{2}\\
&=&\left(-\dfrac{5373}{140}\right)\sqrt{2}+\dfrac{695}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{17}{8}\right)\sqrt{18}\right)-\left(\left(-2\right)\sqrt{50}\right)-\left(\left(\dfrac{33}{5}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{4}+\left(-\dfrac{23}{2}\right)\sqrt{4}+\left(\dfrac{77}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{18}\right)-\left(\left(\left(5\right)\sqrt{50}\right)+\dfrac{25}{4}\right)\right)\times\left(\left(-7\right)\sqrt{8}+\left(\left(\dfrac{21}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{4}\right)+\left(-\dfrac{21}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\left(\dfrac{51}{8}\right)\sqrt{2}\right)-\left(\left(-10\right)\sqrt{2}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{2}\right)\right)-\left(-\dfrac{50}{3}-23+\left(\dfrac{385}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{123}{7}\right)\sqrt{2}\right)-\left(\left(\left(25\right)\sqrt{2}\right)+\dfrac{25}{4}\right)\right)\times\left(\left(-14\right)\sqrt{2}+\left(\left(\dfrac{105}{2}\right)\sqrt{2}\right)-19-\dfrac{11}{2}+\left(-\dfrac{105}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{7333}{140}\right)\sqrt{2}+\dfrac{401}{12}\right)\left(\left(-14\right)\sqrt{2}-\dfrac{49}{2}\right)\\
&=&\left(\dfrac{7333}{10}\right)\sqrt{4}+\left(\dfrac{97853}{120}\right)\sqrt{2}-\dfrac{19649}{24}\\
&=&\dfrac{77747}{120}+\left(\dfrac{97853}{120}\right)\sqrt{2}\\
\end{eqnarray*}