L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{79}{9}\) et \( Y=\left(\left(\left(\dfrac{73}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{50}\right)-\left(\left(-9\right)\sqrt{18}\right)-\dfrac{23}{7}-\left(\left(-\dfrac{23}{2}\right)\sqrt{4}\right)\right)+\dfrac{3}{2}-\left(\left(-\dfrac{4}{3}\right)\sqrt{18}+\dfrac{65}{7}+\left(\dfrac{4}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}+\left(-5\right)\sqrt{4}\right)-\left(\dfrac{77}{4}+\left(-\dfrac{77}{3}\right)\sqrt{50}+\left(-\dfrac{4}{3}\right)\sqrt{18}-\dfrac{38}{9}+\left(-\dfrac{23}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{15}{8}\right)\sqrt{8}+\left(\dfrac{49}{4}\right)\sqrt{4}-\dfrac{3}{2}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{79}{9}\right)+\left(\left(\left(\left(\dfrac{73}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{50}\right)-\left(\left(-9\right)\sqrt{18}\right)-\dfrac{23}{7}-\left(\left(-\dfrac{23}{2}\right)\sqrt{4}\right)\right)+\dfrac{3}{2}-\left(\left(-\dfrac{4}{3}\right)\sqrt{18}+\dfrac{65}{7}+\left(\dfrac{4}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}+\left(-5\right)\sqrt{4}\right)-\left(\dfrac{77}{4}+\left(-\dfrac{77}{3}\right)\sqrt{50}+\left(-\dfrac{4}{3}\right)\sqrt{18}-\dfrac{38}{9}+\left(-\dfrac{23}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{15}{8}\right)\sqrt{8}+\left(\dfrac{49}{4}\right)\sqrt{4}-\dfrac{3}{2}\right)\right)\\
&=&\left(\dfrac{79}{9}\right)+\left(\left(\left(\left(\dfrac{365}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{85}{3}\right)\sqrt{2}\right)-\left(\left(-27\right)\sqrt{2}\right)-\dfrac{23}{7}+23\right)+\dfrac{3}{2}-\left(\left(-4\right)\sqrt{2}+\dfrac{65}{7}+\left(\dfrac{20}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}-10\right)-\left(\dfrac{77}{4}+\left(-\dfrac{385}{3}\right)\sqrt{2}+\left(-4\right)\sqrt{2}-\dfrac{38}{9}-23\right)-\left(\left(-\dfrac{15}{4}\right)\sqrt{2}+\dfrac{49}{2}-\dfrac{3}{2}\right)\right)\\
&=&\dfrac{79}{9}+\left(\left(\left(\dfrac{365}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{85}{3}\right)\sqrt{2}\right)-\left(\left(-27\right)\sqrt{2}\right)-\dfrac{23}{7}+23\right)+\dfrac{3}{2}-\left(\left(-4\right)\sqrt{2}+\dfrac{65}{7}+\left(\dfrac{20}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}-10\right)-\left(\dfrac{77}{4}+\left(-\dfrac{385}{3}\right)\sqrt{2}+\left(-4\right)\sqrt{2}-\dfrac{38}{9}-23\right)-\left(\left(-\dfrac{15}{4}\right)\sqrt{2}+\dfrac{49}{2}-\dfrac{3}{2}\right)\\
&=&\dfrac{439}{28}+\left(\dfrac{49925}{252}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{79}{9}\right)-\left(\left(\left(\left(\dfrac{73}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{50}\right)-\left(\left(-9\right)\sqrt{18}\right)-\dfrac{23}{7}-\left(\left(-\dfrac{23}{2}\right)\sqrt{4}\right)\right)+\dfrac{3}{2}-\left(\left(-\dfrac{4}{3}\right)\sqrt{18}+\dfrac{65}{7}+\left(\dfrac{4}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}+\left(-5\right)\sqrt{4}\right)-\left(\dfrac{77}{4}+\left(-\dfrac{77}{3}\right)\sqrt{50}+\left(-\dfrac{4}{3}\right)\sqrt{18}-\dfrac{38}{9}+\left(-\dfrac{23}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{15}{8}\right)\sqrt{8}+\left(\dfrac{49}{4}\right)\sqrt{4}-\dfrac{3}{2}\right)\right)\\
&=&\left(\dfrac{79}{9}\right)-\left(\left(\left(\left(\dfrac{365}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{85}{3}\right)\sqrt{2}\right)-\left(\left(-27\right)\sqrt{2}\right)-\dfrac{23}{7}+23\right)+\dfrac{3}{2}-\left(\left(-4\right)\sqrt{2}+\dfrac{65}{7}+\left(\dfrac{20}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}-10\right)-\left(\dfrac{77}{4}+\left(-\dfrac{385}{3}\right)\sqrt{2}+\left(-4\right)\sqrt{2}-\dfrac{38}{9}-23\right)-\left(\left(-\dfrac{15}{4}\right)\sqrt{2}+\dfrac{49}{2}-\dfrac{3}{2}\right)\right)\\
&=&\left(\dfrac{79}{9}\right)-\left(\left(\dfrac{49925}{252}\right)\sqrt{2}+\dfrac{1739}{252}\right)\\
&=&\dfrac{79}{9}+\left(-\dfrac{49925}{252}\right)\sqrt{2}-\dfrac{1739}{252}\\
&=&\dfrac{473}{252}+\left(-\dfrac{49925}{252}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{79}{9}\right)\times\left(\left(\left(\left(\dfrac{73}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{50}\right)-\left(\left(-9\right)\sqrt{18}\right)-\dfrac{23}{7}-\left(\left(-\dfrac{23}{2}\right)\sqrt{4}\right)\right)+\dfrac{3}{2}-\left(\left(-\dfrac{4}{3}\right)\sqrt{18}+\dfrac{65}{7}+\left(\dfrac{4}{7}\right)\sqrt{50}+\left(7\right)\sqrt{50}+\left(-5\right)\sqrt{4}\right)-\left(\dfrac{77}{4}+\left(-\dfrac{77}{3}\right)\sqrt{50}+\left(-\dfrac{4}{3}\right)\sqrt{18}-\dfrac{38}{9}+\left(-\dfrac{23}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{15}{8}\right)\sqrt{8}+\left(\dfrac{49}{4}\right)\sqrt{4}-\dfrac{3}{2}\right)\right)\\
&=&\left(\dfrac{79}{9}\right)\times\left(\left(\left(\left(\dfrac{365}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{85}{3}\right)\sqrt{2}\right)-\left(\left(-27\right)\sqrt{2}\right)-\dfrac{23}{7}+23\right)+\dfrac{3}{2}-\left(\left(-4\right)\sqrt{2}+\dfrac{65}{7}+\left(\dfrac{20}{7}\right)\sqrt{2}+\left(35\right)\sqrt{2}-10\right)-\left(\dfrac{77}{4}+\left(-\dfrac{385}{3}\right)\sqrt{2}+\left(-4\right)\sqrt{2}-\dfrac{38}{9}-23\right)-\left(\left(-\dfrac{15}{4}\right)\sqrt{2}+\dfrac{49}{2}-\dfrac{3}{2}\right)\right)\\
&=&\left(\dfrac{79}{9}\right)\left(\left(\dfrac{49925}{252}\right)\sqrt{2}+\dfrac{1739}{252}\right)\\
&=&\left(\dfrac{3944075}{2268}\right)\sqrt{2}+\dfrac{137381}{2268}\\
&=&\left(\dfrac{3944075}{2268}\right)\sqrt{2}+\dfrac{137381}{2268}\\
\end{eqnarray*}