L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{50}{7}\right)\sqrt{8}\) et \( Y=\left(\dfrac{73}{5}\right)\sqrt{4}+\left(\left(\dfrac{25}{8}\right)\sqrt{50}\right)-\left(\left(\dfrac{9}{7}\right)\sqrt{18}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{50}{7}\right)\sqrt{8}\right)+\left(\left(\dfrac{73}{5}\right)\sqrt{4}+\left(\left(\dfrac{25}{8}\right)\sqrt{50}\right)-\left(\left(\dfrac{9}{7}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(-\dfrac{100}{7}\right)\sqrt{2}\right)+\left(\dfrac{146}{5}+\left(\left(\dfrac{125}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{27}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(-\dfrac{100}{7}\right)\sqrt{2}+\dfrac{146}{5}+\left(\left(\dfrac{125}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{27}{7}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{141}{56}\right)\sqrt{2}+\dfrac{146}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{50}{7}\right)\sqrt{8}\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{4}+\left(\left(\dfrac{25}{8}\right)\sqrt{50}\right)-\left(\left(\dfrac{9}{7}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(-\dfrac{100}{7}\right)\sqrt{2}\right)-\left(\dfrac{146}{5}+\left(\left(\dfrac{125}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{27}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{100}{7}\right)\sqrt{2}\right)-\left(\dfrac{146}{5}+\left(\dfrac{659}{56}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{100}{7}\right)\sqrt{2}+-\dfrac{146}{5}+\left(-\dfrac{659}{56}\right)\sqrt{2}\\
&=&\left(-\dfrac{1459}{56}\right)\sqrt{2}-\dfrac{146}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{50}{7}\right)\sqrt{8}\right)\times\left(\left(\dfrac{73}{5}\right)\sqrt{4}+\left(\left(\dfrac{25}{8}\right)\sqrt{50}\right)-\left(\left(\dfrac{9}{7}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(-\dfrac{100}{7}\right)\sqrt{2}\right)\times\left(\dfrac{146}{5}+\left(\left(\dfrac{125}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{27}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{100}{7}\right)\sqrt{2}\right)\left(\dfrac{146}{5}+\left(\dfrac{659}{56}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{2920}{7}\right)\sqrt{2}+\left(-\dfrac{16475}{98}\right)\sqrt{4}\\
&=&\left(-\dfrac{2920}{7}\right)\sqrt{2}-\dfrac{16475}{49}\\
\end{eqnarray*}