L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{76}{5}\right)\sqrt{27}\) et \( Y=\left(-\dfrac{35}{3}\right)\sqrt{12}-\dfrac{80}{7}+\left(-\dfrac{81}{4}\right)\sqrt{27}+\left(\left(-\dfrac{46}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{1}{6}\right)\sqrt{27}\right)+\dfrac{47}{6}-\dfrac{11}{4}-\dfrac{47}{6}\) . 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{76}{5}\right)\sqrt{27}\right)+\left(\left(-\dfrac{35}{3}\right)\sqrt{12}-\dfrac{80}{7}+\left(-\dfrac{81}{4}\right)\sqrt{27}+\left(\left(-\dfrac{46}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{1}{6}\right)\sqrt{27}\right)+\dfrac{47}{6}-\dfrac{11}{4}-\dfrac{47}{6}\right)\\
&=&\left(\left(-\dfrac{228}{5}\right)\sqrt{3}\right)+\left(\left(-\dfrac{70}{3}\right)\sqrt{3}-\dfrac{80}{7}+\left(-\dfrac{243}{4}\right)\sqrt{3}-\dfrac{46}{3}-\left(\left(\dfrac{1}{2}\right)\sqrt{3}\right)+\dfrac{47}{6}-\dfrac{11}{4}-\dfrac{47}{6}\right)\\
&=&\left(-\dfrac{228}{5}\right)\sqrt{3}+\left(-\dfrac{70}{3}\right)\sqrt{3}-\dfrac{80}{7}+\left(-\dfrac{243}{4}\right)\sqrt{3}-\dfrac{46}{3}-\left(\left(\dfrac{1}{2}\right)\sqrt{3}\right)+\dfrac{47}{6}-\dfrac{11}{4}-\dfrac{47}{6}\\
&=&\left(-\dfrac{7811}{60}\right)\sqrt{3}-\dfrac{2479}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{76}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{12}-\dfrac{80}{7}+\left(-\dfrac{81}{4}\right)\sqrt{27}+\left(\left(-\dfrac{46}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{1}{6}\right)\sqrt{27}\right)+\dfrac{47}{6}-\dfrac{11}{4}-\dfrac{47}{6}\right)\\
&=&\left(\left(-\dfrac{228}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{70}{3}\right)\sqrt{3}-\dfrac{80}{7}+\left(-\dfrac{243}{4}\right)\sqrt{3}-\dfrac{46}{3}-\left(\left(\dfrac{1}{2}\right)\sqrt{3}\right)+\dfrac{47}{6}-\dfrac{11}{4}-\dfrac{47}{6}\right)\\
&=&\left(\left(-\dfrac{228}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{1015}{12}\right)\sqrt{3}-\dfrac{2479}{84}\right)\\
&=&\left(-\dfrac{228}{5}\right)\sqrt{3}+\left(\dfrac{1015}{12}\right)\sqrt{3}+\dfrac{2479}{84}\\
&=&\left(\dfrac{2339}{60}\right)\sqrt{3}+\dfrac{2479}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{76}{5}\right)\sqrt{27}\right)\times\left(\left(-\dfrac{35}{3}\right)\sqrt{12}-\dfrac{80}{7}+\left(-\dfrac{81}{4}\right)\sqrt{27}+\left(\left(-\dfrac{46}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{1}{6}\right)\sqrt{27}\right)+\dfrac{47}{6}-\dfrac{11}{4}-\dfrac{47}{6}\right)\\
&=&\left(\left(-\dfrac{228}{5}\right)\sqrt{3}\right)\times\left(\left(-\dfrac{70}{3}\right)\sqrt{3}-\dfrac{80}{7}+\left(-\dfrac{243}{4}\right)\sqrt{3}-\dfrac{46}{3}-\left(\left(\dfrac{1}{2}\right)\sqrt{3}\right)+\dfrac{47}{6}-\dfrac{11}{4}-\dfrac{47}{6}\right)\\
&=&\left(\left(-\dfrac{228}{5}\right)\sqrt{3}\right)\left(\left(-\dfrac{1015}{12}\right)\sqrt{3}-\dfrac{2479}{84}\right)\\
&=&\left(3857\right)\sqrt{9}+\left(\dfrac{47101}{35}\right)\sqrt{3}\\
&=&11571+\left(\dfrac{47101}{35}\right)\sqrt{3}\\
\end{eqnarray*}