L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{17}{4}\right)\sqrt{12}\right)+1-\left(\left(\left(\dfrac{3}{2}\right)\sqrt{27}\right)-\left(\left(-9\right)\sqrt{9}\right)-\left(\left(\dfrac{44}{5}\right)\sqrt{9}\right)\right)\) et \( Y=\left(\left(-\dfrac{81}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{27}+\left(0\right)\sqrt{12}+\left(\dfrac{21}{8}\right)\sqrt{27}+\dfrac{7}{2}\right)-\left(\left(\left(\dfrac{38}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{7}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{75}\right)\right)-\left(-\dfrac{2}{3}+\left(-\dfrac{16}{9}\right)\sqrt{12}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{17}{4}\right)\sqrt{12}\right)+1-\left(\left(\left(\dfrac{3}{2}\right)\sqrt{27}\right)-\left(\left(-9\right)\sqrt{9}\right)-\left(\left(\dfrac{44}{5}\right)\sqrt{9}\right)\right)\right)+\left(\left(\left(-\dfrac{81}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{27}+\left(0\right)\sqrt{12}+\left(\dfrac{21}{8}\right)\sqrt{27}+\dfrac{7}{2}\right)-\left(\left(\left(\dfrac{38}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{7}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{75}\right)\right)-\left(-\dfrac{2}{3}+\left(-\dfrac{16}{9}\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\left(-\dfrac{17}{2}\right)\sqrt{3}\right)+1-\left(\left(\left(\dfrac{9}{2}\right)\sqrt{3}\right)+27-\dfrac{132}{5}\right)\right)+\left(\left(\left(-\dfrac{405}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(\dfrac{63}{8}\right)\sqrt{3}+\dfrac{7}{2}\right)-\left(\left(\left(\dfrac{76}{7}\right)\sqrt{3}\right)-\left(\left(-7\right)\sqrt{3}\right)-\left(\left(\dfrac{375}{7}\right)\sqrt{3}\right)\right)-\left(-\dfrac{2}{3}+\left(-\dfrac{32}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{17}{2}\right)\sqrt{3}\right)+1-\left(\left(\left(\dfrac{9}{2}\right)\sqrt{3}\right)+27-\dfrac{132}{5}\right)+\left(\left(-\dfrac{405}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(\dfrac{63}{8}\right)\sqrt{3}+\dfrac{7}{2}\right)-\left(\left(\left(\dfrac{76}{7}\right)\sqrt{3}\right)-\left(\left(-7\right)\sqrt{3}\right)-\left(\left(\dfrac{375}{7}\right)\sqrt{3}\right)\right)-\left(-\dfrac{2}{3}+\left(-\dfrac{32}{9}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{36359}{504}\right)\sqrt{3}-\dfrac{73}{30}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{17}{4}\right)\sqrt{12}\right)+1-\left(\left(\left(\dfrac{3}{2}\right)\sqrt{27}\right)-\left(\left(-9\right)\sqrt{9}\right)-\left(\left(\dfrac{44}{5}\right)\sqrt{9}\right)\right)\right)-\left(\left(\left(-\dfrac{81}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{27}+\left(0\right)\sqrt{12}+\left(\dfrac{21}{8}\right)\sqrt{27}+\dfrac{7}{2}\right)-\left(\left(\left(\dfrac{38}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{7}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{75}\right)\right)-\left(-\dfrac{2}{3}+\left(-\dfrac{16}{9}\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\left(-\dfrac{17}{2}\right)\sqrt{3}\right)+1-\left(\left(\left(\dfrac{9}{2}\right)\sqrt{3}\right)+27-\dfrac{132}{5}\right)\right)-\left(\left(\left(-\dfrac{405}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(\dfrac{63}{8}\right)\sqrt{3}+\dfrac{7}{2}\right)-\left(\left(\left(\dfrac{76}{7}\right)\sqrt{3}\right)-\left(\left(-7\right)\sqrt{3}\right)-\left(\left(\dfrac{375}{7}\right)\sqrt{3}\right)\right)-\left(-\dfrac{2}{3}+\left(-\dfrac{32}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-13\right)\sqrt{3}+\dfrac{2}{5}\right)-\left(\left(-\dfrac{29807}{504}\right)\sqrt{3}-\dfrac{17}{6}\right)\\
&=&\left(-13\right)\sqrt{3}+\dfrac{2}{5}+\left(\dfrac{29807}{504}\right)\sqrt{3}+\dfrac{17}{6}\\
&=&\left(\dfrac{23255}{504}\right)\sqrt{3}+\dfrac{97}{30}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{17}{4}\right)\sqrt{12}\right)+1-\left(\left(\left(\dfrac{3}{2}\right)\sqrt{27}\right)-\left(\left(-9\right)\sqrt{9}\right)-\left(\left(\dfrac{44}{5}\right)\sqrt{9}\right)\right)\right)\times\left(\left(\left(-\dfrac{81}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{27}+\left(0\right)\sqrt{12}+\left(\dfrac{21}{8}\right)\sqrt{27}+\dfrac{7}{2}\right)-\left(\left(\left(\dfrac{38}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{7}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{75}\right)\right)-\left(-\dfrac{2}{3}+\left(-\dfrac{16}{9}\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\left(-\dfrac{17}{2}\right)\sqrt{3}\right)+1-\left(\left(\left(\dfrac{9}{2}\right)\sqrt{3}\right)+27-\dfrac{132}{5}\right)\right)\times\left(\left(\left(-\dfrac{405}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(\dfrac{63}{8}\right)\sqrt{3}+\dfrac{7}{2}\right)-\left(\left(\left(\dfrac{76}{7}\right)\sqrt{3}\right)-\left(\left(-7\right)\sqrt{3}\right)-\left(\left(\dfrac{375}{7}\right)\sqrt{3}\right)\right)-\left(-\dfrac{2}{3}+\left(-\dfrac{32}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-13\right)\sqrt{3}+\dfrac{2}{5}\right)\left(\left(-\dfrac{29807}{504}\right)\sqrt{3}-\dfrac{17}{6}\right)\\
&=&\left(\dfrac{387491}{504}\right)\sqrt{9}+\left(\dfrac{16603}{1260}\right)\sqrt{3}-\dfrac{17}{15}\\
&=&\dfrac{645501}{280}+\left(\dfrac{16603}{1260}\right)\sqrt{3}\\
\end{eqnarray*}