L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-4-\left(\left(\dfrac{17}{2}\right)\sqrt{18}\right)+\left(7\right)\sqrt{50}\) et \( Y=\left(\dfrac{7}{4}-\left(\left(-\dfrac{40}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{59}{3}\right)\sqrt{18}\right)\right)+5-\left(\left(-\dfrac{20}{9}\right)\sqrt{4}+\left(\dfrac{27}{8}\right)\sqrt{50}+\left(-\dfrac{17}{2}\right)\sqrt{18}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-4-\left(\left(\dfrac{17}{2}\right)\sqrt{18}\right)+\left(7\right)\sqrt{50}\right)+\left(\left(\dfrac{7}{4}-\left(\left(-\dfrac{40}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{59}{3}\right)\sqrt{18}\right)\right)+5-\left(\left(-\dfrac{20}{9}\right)\sqrt{4}+\left(\dfrac{27}{8}\right)\sqrt{50}+\left(-\dfrac{17}{2}\right)\sqrt{18}\right)\right)\\
&=&\left(-4-\left(\left(\dfrac{51}{2}\right)\sqrt{2}\right)+\left(35\right)\sqrt{2}\right)+\left(\left(\dfrac{7}{4}-\left(\left(-\dfrac{200}{7}\right)\sqrt{2}\right)-\left(\left(-59\right)\sqrt{2}\right)\right)+5-\left(-\dfrac{40}{9}+\left(\dfrac{135}{8}\right)\sqrt{2}+\left(-\dfrac{51}{2}\right)\sqrt{2}\right)\right)\\
&=&-4-\left(\left(\dfrac{51}{2}\right)\sqrt{2}\right)+\left(35\right)\sqrt{2}+\left(\dfrac{7}{4}-\left(\left(-\dfrac{200}{7}\right)\sqrt{2}\right)-\left(\left(-59\right)\sqrt{2}\right)\right)+5-\left(-\dfrac{40}{9}+\left(\dfrac{135}{8}\right)\sqrt{2}+\left(-\dfrac{51}{2}\right)\sqrt{2}\right)\\
&=&\dfrac{259}{36}+\left(\dfrac{5919}{56}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-4-\left(\left(\dfrac{17}{2}\right)\sqrt{18}\right)+\left(7\right)\sqrt{50}\right)-\left(\left(\dfrac{7}{4}-\left(\left(-\dfrac{40}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{59}{3}\right)\sqrt{18}\right)\right)+5-\left(\left(-\dfrac{20}{9}\right)\sqrt{4}+\left(\dfrac{27}{8}\right)\sqrt{50}+\left(-\dfrac{17}{2}\right)\sqrt{18}\right)\right)\\
&=&\left(-4-\left(\left(\dfrac{51}{2}\right)\sqrt{2}\right)+\left(35\right)\sqrt{2}\right)-\left(\left(\dfrac{7}{4}-\left(\left(-\dfrac{200}{7}\right)\sqrt{2}\right)-\left(\left(-59\right)\sqrt{2}\right)\right)+5-\left(-\dfrac{40}{9}+\left(\dfrac{135}{8}\right)\sqrt{2}+\left(-\dfrac{51}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(-4+\left(\dfrac{19}{2}\right)\sqrt{2}\right)-\left(\dfrac{403}{36}+\left(\dfrac{5387}{56}\right)\sqrt{2}\right)\\
&=&-4+\left(\dfrac{19}{2}\right)\sqrt{2}+-\dfrac{403}{36}+\left(-\dfrac{5387}{56}\right)\sqrt{2}\\
&=&-\dfrac{547}{36}+\left(-\dfrac{4855}{56}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-4-\left(\left(\dfrac{17}{2}\right)\sqrt{18}\right)+\left(7\right)\sqrt{50}\right)\times\left(\left(\dfrac{7}{4}-\left(\left(-\dfrac{40}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{59}{3}\right)\sqrt{18}\right)\right)+5-\left(\left(-\dfrac{20}{9}\right)\sqrt{4}+\left(\dfrac{27}{8}\right)\sqrt{50}+\left(-\dfrac{17}{2}\right)\sqrt{18}\right)\right)\\
&=&\left(-4-\left(\left(\dfrac{51}{2}\right)\sqrt{2}\right)+\left(35\right)\sqrt{2}\right)\times\left(\left(\dfrac{7}{4}-\left(\left(-\dfrac{200}{7}\right)\sqrt{2}\right)-\left(\left(-59\right)\sqrt{2}\right)\right)+5-\left(-\dfrac{40}{9}+\left(\dfrac{135}{8}\right)\sqrt{2}+\left(-\dfrac{51}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(-4+\left(\dfrac{19}{2}\right)\sqrt{2}\right)\left(\dfrac{403}{36}+\left(\dfrac{5387}{56}\right)\sqrt{2}\right)\\
&=&-\dfrac{403}{9}+\left(-\dfrac{140333}{504}\right)\sqrt{2}+\left(\dfrac{102353}{112}\right)\sqrt{4}\\
&=&\dfrac{898609}{504}+\left(-\dfrac{140333}{504}\right)\sqrt{2}\\
\end{eqnarray*}