L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{1}{4}-\dfrac{21}{5}+\left(\left(6\right)\sqrt{8}\right)+2-\left(\left(-\dfrac{7}{8}\right)\sqrt{18}\right)\) et \( Y=-\dfrac{1}{5}-\left(\left(\left(\dfrac{7}{2}\right)\sqrt{8}\right)-\dfrac{1}{3}-\left(\left(1\right)\sqrt{50}\right)-\left(\left(0\right)\sqrt{18}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{18}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{1}{4}-\dfrac{21}{5}+\left(\left(6\right)\sqrt{8}\right)+2-\left(\left(-\dfrac{7}{8}\right)\sqrt{18}\right)\right)+\left(-\dfrac{1}{5}-\left(\left(\left(\dfrac{7}{2}\right)\sqrt{8}\right)-\dfrac{1}{3}-\left(\left(1\right)\sqrt{50}\right)-\left(\left(0\right)\sqrt{18}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\dfrac{1}{4}-\dfrac{21}{5}+\left(\left(12\right)\sqrt{2}\right)+2-\left(\left(-\dfrac{21}{8}\right)\sqrt{2}\right)\right)+\left(-\dfrac{1}{5}-\left(\left(\left(7\right)\sqrt{2}\right)-\dfrac{1}{3}-\left(\left(5\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{135}{8}\right)\sqrt{2}\right)\right)\right)\\
&=&\dfrac{1}{4}-\dfrac{21}{5}+\left(\left(12\right)\sqrt{2}\right)+2-\left(\left(-\dfrac{21}{8}\right)\sqrt{2}\right)-\dfrac{1}{5}-\left(\left(\left(7\right)\sqrt{2}\right)-\dfrac{1}{3}-\left(\left(5\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{135}{8}\right)\sqrt{2}\right)\right)\\
&=&-\dfrac{109}{60}+\left(\dfrac{59}{2}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{1}{4}-\dfrac{21}{5}+\left(\left(6\right)\sqrt{8}\right)+2-\left(\left(-\dfrac{7}{8}\right)\sqrt{18}\right)\right)-\left(-\dfrac{1}{5}-\left(\left(\left(\dfrac{7}{2}\right)\sqrt{8}\right)-\dfrac{1}{3}-\left(\left(1\right)\sqrt{50}\right)-\left(\left(0\right)\sqrt{18}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\dfrac{1}{4}-\dfrac{21}{5}+\left(\left(12\right)\sqrt{2}\right)+2-\left(\left(-\dfrac{21}{8}\right)\sqrt{2}\right)\right)-\left(-\dfrac{1}{5}-\left(\left(\left(7\right)\sqrt{2}\right)-\dfrac{1}{3}-\left(\left(5\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{135}{8}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(-\dfrac{39}{20}+\left(\dfrac{117}{8}\right)\sqrt{2}\right)-\left(\dfrac{2}{15}+\left(\dfrac{119}{8}\right)\sqrt{2}\right)\\
&=&-\dfrac{39}{20}+\left(\dfrac{117}{8}\right)\sqrt{2}+-\dfrac{2}{15}+\left(-\dfrac{119}{8}\right)\sqrt{2}\\
&=&-\dfrac{25}{12}+\left(-\dfrac{1}{4}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{1}{4}-\dfrac{21}{5}+\left(\left(6\right)\sqrt{8}\right)+2-\left(\left(-\dfrac{7}{8}\right)\sqrt{18}\right)\right)\times\left(-\dfrac{1}{5}-\left(\left(\left(\dfrac{7}{2}\right)\sqrt{8}\right)-\dfrac{1}{3}-\left(\left(1\right)\sqrt{50}\right)-\left(\left(0\right)\sqrt{18}\right)-\left(\left(\dfrac{45}{8}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\dfrac{1}{4}-\dfrac{21}{5}+\left(\left(12\right)\sqrt{2}\right)+2-\left(\left(-\dfrac{21}{8}\right)\sqrt{2}\right)\right)\times\left(-\dfrac{1}{5}-\left(\left(\left(7\right)\sqrt{2}\right)-\dfrac{1}{3}-\left(\left(5\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{135}{8}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(-\dfrac{39}{20}+\left(\dfrac{117}{8}\right)\sqrt{2}\right)\left(\dfrac{2}{15}+\left(\dfrac{119}{8}\right)\sqrt{2}\right)\\
&=&-\dfrac{13}{50}+\left(-\dfrac{4329}{160}\right)\sqrt{2}+\left(\dfrac{13923}{64}\right)\sqrt{4}\\
&=&\dfrac{347867}{800}+\left(-\dfrac{4329}{160}\right)\sqrt{2}\\
\end{eqnarray*}