L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(\left(8\right)\sqrt{12}\right)-\dfrac{19}{3}-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{59}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{62}{5}\right)\sqrt{9}\right)\right)\) et \( Y=\dfrac{37}{2}-7+\left(-\dfrac{77}{4}\right)\sqrt{9}+\left(\dfrac{43}{8}\right)\sqrt{12}+\left(\dfrac{33}{2}\right)\sqrt{12}+\left(\dfrac{2}{5}\right)\sqrt{12}+\dfrac{46}{5}\) . 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{7}{2}\right)\sqrt{27}\right)-\left(\left(\left(8\right)\sqrt{12}\right)-\dfrac{19}{3}-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{59}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{62}{5}\right)\sqrt{9}\right)\right)\right)+\left(\dfrac{37}{2}-7+\left(-\dfrac{77}{4}\right)\sqrt{9}+\left(\dfrac{43}{8}\right)\sqrt{12}+\left(\dfrac{33}{2}\right)\sqrt{12}+\left(\dfrac{2}{5}\right)\sqrt{12}+\dfrac{46}{5}\right)\\
&=&\left(\left(\left(-\dfrac{21}{2}\right)\sqrt{3}\right)-\left(\left(\left(16\right)\sqrt{3}\right)-\dfrac{19}{3}-\left(\left(-25\right)\sqrt{3}\right)-\left(\left(\dfrac{59}{3}\right)\sqrt{3}\right)+\dfrac{186}{5}\right)\right)+\left(\dfrac{37}{2}-7-\dfrac{231}{4}+\left(\dfrac{43}{4}\right)\sqrt{3}+\left(33\right)\sqrt{3}+\left(\dfrac{4}{5}\right)\sqrt{3}+\dfrac{46}{5}\right)\\
&=&\left(\left(-\dfrac{21}{2}\right)\sqrt{3}\right)-\left(\left(\left(16\right)\sqrt{3}\right)-\dfrac{19}{3}-\left(\left(-25\right)\sqrt{3}\right)-\left(\left(\dfrac{59}{3}\right)\sqrt{3}\right)+\dfrac{186}{5}\right)+\dfrac{37}{2}-7-\dfrac{231}{4}+\left(\dfrac{43}{4}\right)\sqrt{3}+\left(33\right)\sqrt{3}+\left(\dfrac{4}{5}\right)\sqrt{3}+\dfrac{46}{5}\\
&=&\left(\dfrac{763}{60}\right)\sqrt{3}-\dfrac{815}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(\left(8\right)\sqrt{12}\right)-\dfrac{19}{3}-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{59}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{62}{5}\right)\sqrt{9}\right)\right)\right)-\left(\dfrac{37}{2}-7+\left(-\dfrac{77}{4}\right)\sqrt{9}+\left(\dfrac{43}{8}\right)\sqrt{12}+\left(\dfrac{33}{2}\right)\sqrt{12}+\left(\dfrac{2}{5}\right)\sqrt{12}+\dfrac{46}{5}\right)\\
&=&\left(\left(\left(-\dfrac{21}{2}\right)\sqrt{3}\right)-\left(\left(\left(16\right)\sqrt{3}\right)-\dfrac{19}{3}-\left(\left(-25\right)\sqrt{3}\right)-\left(\left(\dfrac{59}{3}\right)\sqrt{3}\right)+\dfrac{186}{5}\right)\right)-\left(\dfrac{37}{2}-7-\dfrac{231}{4}+\left(\dfrac{43}{4}\right)\sqrt{3}+\left(33\right)\sqrt{3}+\left(\dfrac{4}{5}\right)\sqrt{3}+\dfrac{46}{5}\right)\\
&=&\left(\left(-\dfrac{191}{6}\right)\sqrt{3}-\dfrac{463}{15}\right)-\left(-\dfrac{741}{20}+\left(\dfrac{891}{20}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{191}{6}\right)\sqrt{3}-\dfrac{463}{15}+\dfrac{741}{20}+\left(-\dfrac{891}{20}\right)\sqrt{3}\\
&=&\left(-\dfrac{4583}{60}\right)\sqrt{3}+\dfrac{371}{60}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(\left(8\right)\sqrt{12}\right)-\dfrac{19}{3}-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{59}{9}\right)\sqrt{27}\right)-\left(\left(-\dfrac{62}{5}\right)\sqrt{9}\right)\right)\right)\times\left(\dfrac{37}{2}-7+\left(-\dfrac{77}{4}\right)\sqrt{9}+\left(\dfrac{43}{8}\right)\sqrt{12}+\left(\dfrac{33}{2}\right)\sqrt{12}+\left(\dfrac{2}{5}\right)\sqrt{12}+\dfrac{46}{5}\right)\\
&=&\left(\left(\left(-\dfrac{21}{2}\right)\sqrt{3}\right)-\left(\left(\left(16\right)\sqrt{3}\right)-\dfrac{19}{3}-\left(\left(-25\right)\sqrt{3}\right)-\left(\left(\dfrac{59}{3}\right)\sqrt{3}\right)+\dfrac{186}{5}\right)\right)\times\left(\dfrac{37}{2}-7-\dfrac{231}{4}+\left(\dfrac{43}{4}\right)\sqrt{3}+\left(33\right)\sqrt{3}+\left(\dfrac{4}{5}\right)\sqrt{3}+\dfrac{46}{5}\right)\\
&=&\left(\left(-\dfrac{191}{6}\right)\sqrt{3}-\dfrac{463}{15}\right)\left(-\dfrac{741}{20}+\left(\dfrac{891}{20}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{39137}{200}\right)\sqrt{3}+\left(-\dfrac{56727}{40}\right)\sqrt{9}+\dfrac{114361}{100}\\
&=&\left(-\dfrac{39137}{200}\right)\sqrt{3}-\dfrac{622183}{200}\\
\end{eqnarray*}