L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(-\dfrac{19}{9}\right)\sqrt{50}\) et \( Y=\left(\left(-\dfrac{1}{2}\right)\sqrt{4}+\left(\dfrac{35}{6}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{8}\right)+3\right)-\left(-\dfrac{67}{5}+6-\left(\left(-5\right)\sqrt{8}\right)+\dfrac{17}{6}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{8}-\dfrac{57}{8}+\dfrac{49}{4}+\left(\dfrac{38}{9}\right)\sqrt{50}+\left(-5\right)\sqrt{18}\right)-\left(\left(-\dfrac{2}{9}\right)\sqrt{18}+\left(\dfrac{77}{4}\right)\sqrt{18}+\left(-\dfrac{43}{2}\right)\sqrt{8}+\left(-5\right)\sqrt{8}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{19}{9}\right)\sqrt{50}\right)+\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{4}+\left(\dfrac{35}{6}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{8}\right)+3\right)-\left(-\dfrac{67}{5}+6-\left(\left(-5\right)\sqrt{8}\right)+\dfrac{17}{6}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{8}-\dfrac{57}{8}+\dfrac{49}{4}+\left(\dfrac{38}{9}\right)\sqrt{50}+\left(-5\right)\sqrt{18}\right)-\left(\left(-\dfrac{2}{9}\right)\sqrt{18}+\left(\dfrac{77}{4}\right)\sqrt{18}+\left(-\dfrac{43}{2}\right)\sqrt{8}+\left(-5\right)\sqrt{8}\right)\right)\\
&=&\left(\left(-\dfrac{95}{9}\right)\sqrt{2}\right)+\left(\left(-1+\left(\dfrac{35}{3}\right)\sqrt{2}\right)-\left(\left(\left(-43\right)\sqrt{2}\right)+3\right)-\left(-\dfrac{67}{5}+6-\left(\left(-10\right)\sqrt{2}\right)+\dfrac{17}{6}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{2}-\dfrac{57}{8}+\dfrac{49}{4}+\left(\dfrac{190}{9}\right)\sqrt{2}+\left(-15\right)\sqrt{2}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{2}+\left(\dfrac{231}{4}\right)\sqrt{2}+\left(-43\right)\sqrt{2}+\left(-10\right)\sqrt{2}\right)\right)\\
&=&\left(-\dfrac{95}{9}\right)\sqrt{2}+\left(-1+\left(\dfrac{35}{3}\right)\sqrt{2}\right)-\left(\left(\left(-43\right)\sqrt{2}\right)+3\right)-\left(-\dfrac{67}{5}+6-\left(\left(-10\right)\sqrt{2}\right)+\dfrac{17}{6}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{2}-\dfrac{57}{8}+\dfrac{49}{4}+\left(\dfrac{190}{9}\right)\sqrt{2}+\left(-15\right)\sqrt{2}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{2}+\left(\dfrac{231}{4}\right)\sqrt{2}+\left(-43\right)\sqrt{2}+\left(-10\right)\sqrt{2}\right)\\
&=&\left(\dfrac{293}{12}\right)\sqrt{2}-\dfrac{547}{120}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{19}{9}\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{4}+\left(\dfrac{35}{6}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{8}\right)+3\right)-\left(-\dfrac{67}{5}+6-\left(\left(-5\right)\sqrt{8}\right)+\dfrac{17}{6}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{8}-\dfrac{57}{8}+\dfrac{49}{4}+\left(\dfrac{38}{9}\right)\sqrt{50}+\left(-5\right)\sqrt{18}\right)-\left(\left(-\dfrac{2}{9}\right)\sqrt{18}+\left(\dfrac{77}{4}\right)\sqrt{18}+\left(-\dfrac{43}{2}\right)\sqrt{8}+\left(-5\right)\sqrt{8}\right)\right)\\
&=&\left(\left(-\dfrac{95}{9}\right)\sqrt{2}\right)-\left(\left(-1+\left(\dfrac{35}{3}\right)\sqrt{2}\right)-\left(\left(\left(-43\right)\sqrt{2}\right)+3\right)-\left(-\dfrac{67}{5}+6-\left(\left(-10\right)\sqrt{2}\right)+\dfrac{17}{6}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{2}-\dfrac{57}{8}+\dfrac{49}{4}+\left(\dfrac{190}{9}\right)\sqrt{2}+\left(-15\right)\sqrt{2}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{2}+\left(\dfrac{231}{4}\right)\sqrt{2}+\left(-43\right)\sqrt{2}+\left(-10\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{95}{9}\right)\sqrt{2}\right)-\left(-\dfrac{547}{120}+\left(\dfrac{1259}{36}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{95}{9}\right)\sqrt{2}+\dfrac{547}{120}+\left(-\dfrac{1259}{36}\right)\sqrt{2}\\
&=&\left(-\dfrac{1639}{36}\right)\sqrt{2}+\dfrac{547}{120}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{19}{9}\right)\sqrt{50}\right)\times\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{4}+\left(\dfrac{35}{6}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{8}\right)+3\right)-\left(-\dfrac{67}{5}+6-\left(\left(-5\right)\sqrt{8}\right)+\dfrac{17}{6}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{8}-\dfrac{57}{8}+\dfrac{49}{4}+\left(\dfrac{38}{9}\right)\sqrt{50}+\left(-5\right)\sqrt{18}\right)-\left(\left(-\dfrac{2}{9}\right)\sqrt{18}+\left(\dfrac{77}{4}\right)\sqrt{18}+\left(-\dfrac{43}{2}\right)\sqrt{8}+\left(-5\right)\sqrt{8}\right)\right)\\
&=&\left(\left(-\dfrac{95}{9}\right)\sqrt{2}\right)\times\left(\left(-1+\left(\dfrac{35}{3}\right)\sqrt{2}\right)-\left(\left(\left(-43\right)\sqrt{2}\right)+3\right)-\left(-\dfrac{67}{5}+6-\left(\left(-10\right)\sqrt{2}\right)+\dfrac{17}{6}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{2}-\dfrac{57}{8}+\dfrac{49}{4}+\left(\dfrac{190}{9}\right)\sqrt{2}+\left(-15\right)\sqrt{2}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{2}+\left(\dfrac{231}{4}\right)\sqrt{2}+\left(-43\right)\sqrt{2}+\left(-10\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{95}{9}\right)\sqrt{2}\right)\left(-\dfrac{547}{120}+\left(\dfrac{1259}{36}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{10393}{216}\right)\sqrt{2}+\left(-\dfrac{119605}{324}\right)\sqrt{4}\\
&=&\left(\dfrac{10393}{216}\right)\sqrt{2}-\dfrac{119605}{162}\\
\end{eqnarray*}