L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{40}{9}\) et \( Y=\left(\left(\left(-\dfrac{2}{3}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)+\dfrac{29}{4}\right)-\left(\left(-\dfrac{71}{5}\right)\sqrt{75}+\left(-\dfrac{9}{2}\right)\sqrt{27}+\left(\dfrac{61}{7}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)-\left(\left(-\dfrac{13}{3}\right)\sqrt{9}\right)+4\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{40}{9}\right)+\left(\left(\left(\left(-\dfrac{2}{3}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)+\dfrac{29}{4}\right)-\left(\left(-\dfrac{71}{5}\right)\sqrt{75}+\left(-\dfrac{9}{2}\right)\sqrt{27}+\left(\dfrac{61}{7}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)-\left(\left(-\dfrac{13}{3}\right)\sqrt{9}\right)+4\right)\right)\\
&=&\left(-\dfrac{40}{9}\right)+\left(\left(\left(\left(-\dfrac{4}{3}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)+\dfrac{29}{4}\right)-\left(\left(-71\right)\sqrt{3}+\left(-\dfrac{27}{2}\right)\sqrt{3}+\left(\dfrac{305}{7}\right)\sqrt{3}\right)-\left(\left(\left(-43\right)\sqrt{3}\right)+13+4\right)\right)\\
&=&-\dfrac{40}{9}+\left(\left(\left(-\dfrac{4}{3}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)+\dfrac{29}{4}\right)-\left(\left(-71\right)\sqrt{3}+\left(-\dfrac{27}{2}\right)\sqrt{3}+\left(\dfrac{305}{7}\right)\sqrt{3}\right)-\left(\left(\left(-43\right)\sqrt{3}\right)+13+4\right)\\
&=&-\dfrac{511}{36}+\left(\dfrac{4351}{42}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{40}{9}\right)-\left(\left(\left(\left(-\dfrac{2}{3}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)+\dfrac{29}{4}\right)-\left(\left(-\dfrac{71}{5}\right)\sqrt{75}+\left(-\dfrac{9}{2}\right)\sqrt{27}+\left(\dfrac{61}{7}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)-\left(\left(-\dfrac{13}{3}\right)\sqrt{9}\right)+4\right)\right)\\
&=&\left(-\dfrac{40}{9}\right)-\left(\left(\left(\left(-\dfrac{4}{3}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)+\dfrac{29}{4}\right)-\left(\left(-71\right)\sqrt{3}+\left(-\dfrac{27}{2}\right)\sqrt{3}+\left(\dfrac{305}{7}\right)\sqrt{3}\right)-\left(\left(\left(-43\right)\sqrt{3}\right)+13+4\right)\right)\\
&=&\left(-\dfrac{40}{9}\right)-\left(\left(\dfrac{4351}{42}\right)\sqrt{3}-\dfrac{39}{4}\right)\\
&=&-\dfrac{40}{9}+\left(-\dfrac{4351}{42}\right)\sqrt{3}+\dfrac{39}{4}\\
&=&\dfrac{191}{36}+\left(-\dfrac{4351}{42}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{40}{9}\right)\times\left(\left(\left(\left(-\dfrac{2}{3}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)+\dfrac{29}{4}\right)-\left(\left(-\dfrac{71}{5}\right)\sqrt{75}+\left(-\dfrac{9}{2}\right)\sqrt{27}+\left(\dfrac{61}{7}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)-\left(\left(-\dfrac{13}{3}\right)\sqrt{9}\right)+4\right)\right)\\
&=&\left(-\dfrac{40}{9}\right)\times\left(\left(\left(\left(-\dfrac{4}{3}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)+\dfrac{29}{4}\right)-\left(\left(-71\right)\sqrt{3}+\left(-\dfrac{27}{2}\right)\sqrt{3}+\left(\dfrac{305}{7}\right)\sqrt{3}\right)-\left(\left(\left(-43\right)\sqrt{3}\right)+13+4\right)\right)\\
&=&\left(-\dfrac{40}{9}\right)\left(\left(\dfrac{4351}{42}\right)\sqrt{3}-\dfrac{39}{4}\right)\\
&=&\left(-\dfrac{87020}{189}\right)\sqrt{3}+\dfrac{130}{3}\\
&=&\left(-\dfrac{87020}{189}\right)\sqrt{3}+\dfrac{130}{3}\\
\end{eqnarray*}