L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(8\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{32}{9}\right)\sqrt{50}\right)+\dfrac{17}{9}+\dfrac{17}{8}\right)-\left(\left(\left(-\dfrac{49}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{8}\right)-\left(\left(-\dfrac{23}{8}\right)\sqrt{4}\right)+4\right)\) et \( Y=\left(-\dfrac{19}{4}\right)\sqrt{50}+\dfrac{22}{3}+\dfrac{12}{7}+\dfrac{64}{9}\) . 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(8\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{32}{9}\right)\sqrt{50}\right)+\dfrac{17}{9}+\dfrac{17}{8}\right)-\left(\left(\left(-\dfrac{49}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{8}\right)-\left(\left(-\dfrac{23}{8}\right)\sqrt{4}\right)+4\right)\right)+\left(\left(-\dfrac{19}{4}\right)\sqrt{50}+\dfrac{22}{3}+\dfrac{12}{7}+\dfrac{64}{9}\right)\\
&=&\left(\left(\left(40\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{160}{9}\right)\sqrt{2}\right)+\dfrac{17}{9}+\dfrac{17}{8}\right)-\left(\left(\left(-\dfrac{49}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{56}{5}\right)\sqrt{2}\right)+\dfrac{23}{4}+4\right)\right)+\left(\left(-\dfrac{95}{4}\right)\sqrt{2}+\dfrac{22}{3}+\dfrac{12}{7}+\dfrac{64}{9}\right)\\
&=&\left(\left(40\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{160}{9}\right)\sqrt{2}\right)+\dfrac{17}{9}+\dfrac{17}{8}\right)-\left(\left(\left(-\dfrac{49}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{56}{5}\right)\sqrt{2}\right)+\dfrac{23}{4}+4\right)+\left(-\dfrac{95}{4}\right)\sqrt{2}+\dfrac{22}{3}+\dfrac{12}{7}+\dfrac{64}{9}\\
&=&\left(\dfrac{7049}{180}\right)\sqrt{2}+\dfrac{1207}{504}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(8\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{32}{9}\right)\sqrt{50}\right)+\dfrac{17}{9}+\dfrac{17}{8}\right)-\left(\left(\left(-\dfrac{49}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{8}\right)-\left(\left(-\dfrac{23}{8}\right)\sqrt{4}\right)+4\right)\right)-\left(\left(-\dfrac{19}{4}\right)\sqrt{50}+\dfrac{22}{3}+\dfrac{12}{7}+\dfrac{64}{9}\right)\\
&=&\left(\left(\left(40\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{160}{9}\right)\sqrt{2}\right)+\dfrac{17}{9}+\dfrac{17}{8}\right)-\left(\left(\left(-\dfrac{49}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{56}{5}\right)\sqrt{2}\right)+\dfrac{23}{4}+4\right)\right)-\left(\left(-\dfrac{95}{4}\right)\sqrt{2}+\dfrac{22}{3}+\dfrac{12}{7}+\dfrac{64}{9}\right)\\
&=&\left(\left(\dfrac{2831}{45}\right)\sqrt{2}-\dfrac{991}{72}\right)-\left(\left(-\dfrac{95}{4}\right)\sqrt{2}+\dfrac{1018}{63}\right)\\
&=&\left(\dfrac{2831}{45}\right)\sqrt{2}-\dfrac{991}{72}+\left(\dfrac{95}{4}\right)\sqrt{2}-\dfrac{1018}{63}\\
&=&\left(\dfrac{15599}{180}\right)\sqrt{2}-\dfrac{5027}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(8\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{32}{9}\right)\sqrt{50}\right)+\dfrac{17}{9}+\dfrac{17}{8}\right)-\left(\left(\left(-\dfrac{49}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{8}\right)-\left(\left(-\dfrac{23}{8}\right)\sqrt{4}\right)+4\right)\right)\times\left(\left(-\dfrac{19}{4}\right)\sqrt{50}+\dfrac{22}{3}+\dfrac{12}{7}+\dfrac{64}{9}\right)\\
&=&\left(\left(\left(40\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{160}{9}\right)\sqrt{2}\right)+\dfrac{17}{9}+\dfrac{17}{8}\right)-\left(\left(\left(-\dfrac{49}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{56}{5}\right)\sqrt{2}\right)+\dfrac{23}{4}+4\right)\right)\times\left(\left(-\dfrac{95}{4}\right)\sqrt{2}+\dfrac{22}{3}+\dfrac{12}{7}+\dfrac{64}{9}\right)\\
&=&\left(\left(\dfrac{2831}{45}\right)\sqrt{2}-\dfrac{991}{72}\right)\left(\left(-\dfrac{95}{4}\right)\sqrt{2}+\dfrac{1018}{63}\right)\\
&=&\left(-\dfrac{53789}{36}\right)\sqrt{4}+\left(\dfrac{121878331}{90720}\right)\sqrt{2}-\dfrac{504419}{2268}\\
&=&-\dfrac{7281833}{2268}+\left(\dfrac{121878331}{90720}\right)\sqrt{2}\\
\end{eqnarray*}