L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{36}{5}\right)\sqrt{9}+\left(\dfrac{9}{2}\right)\sqrt{27}+\left(-\dfrac{19}{7}\right)\sqrt{12}+\left(0\right)\sqrt{12}+\left(0\right)\sqrt{27}+\left(-\dfrac{47}{8}\right)\sqrt{27}+\left(-8\right)\sqrt{12}+\left(\dfrac{27}{4}\right)\sqrt{12}\) et \( Y=\left(-\dfrac{17}{4}\right)\sqrt{75}\) . 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{36}{5}\right)\sqrt{9}+\left(\dfrac{9}{2}\right)\sqrt{27}+\left(-\dfrac{19}{7}\right)\sqrt{12}+\left(0\right)\sqrt{12}+\left(0\right)\sqrt{27}+\left(-\dfrac{47}{8}\right)\sqrt{27}+\left(-8\right)\sqrt{12}+\left(\dfrac{27}{4}\right)\sqrt{12}\right)+\left(\left(-\dfrac{17}{4}\right)\sqrt{75}\right)\\
&=&\left(\dfrac{108}{5}+\left(\dfrac{27}{2}\right)\sqrt{3}+\left(-\dfrac{38}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(-\dfrac{141}{8}\right)\sqrt{3}+\left(-16\right)\sqrt{3}+\left(\dfrac{27}{2}\right)\sqrt{3}\right)+\left(\left(-\dfrac{85}{4}\right)\sqrt{3}\right)\\
&=&\dfrac{108}{5}+\left(\dfrac{27}{2}\right)\sqrt{3}+\left(-\dfrac{38}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(-\dfrac{141}{8}\right)\sqrt{3}+\left(-16\right)\sqrt{3}+\left(\dfrac{27}{2}\right)\sqrt{3}+\left(-\dfrac{85}{4}\right)\sqrt{3}\\
&=&\dfrac{108}{5}+\left(-\dfrac{1865}{56}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{36}{5}\right)\sqrt{9}+\left(\dfrac{9}{2}\right)\sqrt{27}+\left(-\dfrac{19}{7}\right)\sqrt{12}+\left(0\right)\sqrt{12}+\left(0\right)\sqrt{27}+\left(-\dfrac{47}{8}\right)\sqrt{27}+\left(-8\right)\sqrt{12}+\left(\dfrac{27}{4}\right)\sqrt{12}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{75}\right)\\
&=&\left(\dfrac{108}{5}+\left(\dfrac{27}{2}\right)\sqrt{3}+\left(-\dfrac{38}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(-\dfrac{141}{8}\right)\sqrt{3}+\left(-16\right)\sqrt{3}+\left(\dfrac{27}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{85}{4}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{108}{5}+\left(-\dfrac{675}{56}\right)\sqrt{3}\right)-\left(\left(-\dfrac{85}{4}\right)\sqrt{3}\right)\\
&=&\dfrac{108}{5}+\left(-\dfrac{675}{56}\right)\sqrt{3}+\left(\dfrac{85}{4}\right)\sqrt{3}\\
&=&\dfrac{108}{5}+\left(\dfrac{515}{56}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{36}{5}\right)\sqrt{9}+\left(\dfrac{9}{2}\right)\sqrt{27}+\left(-\dfrac{19}{7}\right)\sqrt{12}+\left(0\right)\sqrt{12}+\left(0\right)\sqrt{27}+\left(-\dfrac{47}{8}\right)\sqrt{27}+\left(-8\right)\sqrt{12}+\left(\dfrac{27}{4}\right)\sqrt{12}\right)\times\left(\left(-\dfrac{17}{4}\right)\sqrt{75}\right)\\
&=&\left(\dfrac{108}{5}+\left(\dfrac{27}{2}\right)\sqrt{3}+\left(-\dfrac{38}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(-\dfrac{141}{8}\right)\sqrt{3}+\left(-16\right)\sqrt{3}+\left(\dfrac{27}{2}\right)\sqrt{3}\right)\times\left(\left(-\dfrac{85}{4}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{108}{5}+\left(-\dfrac{675}{56}\right)\sqrt{3}\right)\left(\left(-\dfrac{85}{4}\right)\sqrt{3}\right)\\
&=&\left(-459\right)\sqrt{3}+\left(\dfrac{57375}{224}\right)\sqrt{9}\\
&=&\left(-459\right)\sqrt{3}+\dfrac{172125}{224}\\
\end{eqnarray*}