L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{3}{5}\right)\sqrt{12}-\dfrac{27}{5}+\left(-\dfrac{5}{7}\right)\sqrt{27}+\left(\left(\dfrac{34}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{3}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{75}\right)\) et \( Y=\left(\left(-\dfrac{5}{2}\right)\sqrt{75}+\left(\dfrac{1}{2}\right)\sqrt{75}+\left(-9\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{75}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{3}{5}\right)\sqrt{12}-\dfrac{27}{5}+\left(-\dfrac{5}{7}\right)\sqrt{27}+\left(\left(\dfrac{34}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{3}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{75}\right)\right)+\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{75}+\left(\dfrac{1}{2}\right)\sqrt{75}+\left(-9\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\dfrac{6}{5}\right)\sqrt{3}-\dfrac{27}{5}+\left(-\dfrac{15}{7}\right)\sqrt{3}+\left(\left(34\right)\sqrt{3}\right)-\left(\left(\dfrac{325}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{3}\right)\right)+\left(\left(\left(-\dfrac{25}{2}\right)\sqrt{3}+\left(\dfrac{5}{2}\right)\sqrt{3}-27\right)-\left(\left(15\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{6}{5}\right)\sqrt{3}-\dfrac{27}{5}+\left(-\dfrac{15}{7}\right)\sqrt{3}+\left(\left(34\right)\sqrt{3}\right)-\left(\left(\dfrac{325}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{3}\right)+\left(\left(-\dfrac{25}{2}\right)\sqrt{3}+\left(\dfrac{5}{2}\right)\sqrt{3}-27\right)-\left(\left(15\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{359}{14}\right)\sqrt{3}-\dfrac{162}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{3}{5}\right)\sqrt{12}-\dfrac{27}{5}+\left(-\dfrac{5}{7}\right)\sqrt{27}+\left(\left(\dfrac{34}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{3}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{75}\right)\right)-\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{75}+\left(\dfrac{1}{2}\right)\sqrt{75}+\left(-9\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\dfrac{6}{5}\right)\sqrt{3}-\dfrac{27}{5}+\left(-\dfrac{15}{7}\right)\sqrt{3}+\left(\left(34\right)\sqrt{3}\right)-\left(\left(\dfrac{325}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{3}\right)\right)-\left(\left(\left(-\dfrac{25}{2}\right)\sqrt{3}+\left(\dfrac{5}{2}\right)\sqrt{3}-27\right)-\left(\left(15\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{9}{14}\right)\sqrt{3}-\dfrac{27}{5}\right)-\left(\left(-25\right)\sqrt{3}-27\right)\\
&=&\left(-\dfrac{9}{14}\right)\sqrt{3}-\dfrac{27}{5}+\left(25\right)\sqrt{3}+27\\
&=&\left(\dfrac{341}{14}\right)\sqrt{3}+\dfrac{108}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{3}{5}\right)\sqrt{12}-\dfrac{27}{5}+\left(-\dfrac{5}{7}\right)\sqrt{27}+\left(\left(\dfrac{34}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{3}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{75}\right)\right)\times\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{75}+\left(\dfrac{1}{2}\right)\sqrt{75}+\left(-9\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\dfrac{6}{5}\right)\sqrt{3}-\dfrac{27}{5}+\left(-\dfrac{15}{7}\right)\sqrt{3}+\left(\left(34\right)\sqrt{3}\right)-\left(\left(\dfrac{325}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{3}\right)\right)\times\left(\left(\left(-\dfrac{25}{2}\right)\sqrt{3}+\left(\dfrac{5}{2}\right)\sqrt{3}-27\right)-\left(\left(15\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{9}{14}\right)\sqrt{3}-\dfrac{27}{5}\right)\left(\left(-25\right)\sqrt{3}-27\right)\\
&=&\left(\dfrac{225}{14}\right)\sqrt{9}+\left(\dfrac{2133}{14}\right)\sqrt{3}+\dfrac{729}{5}\\
&=&\dfrac{13581}{70}+\left(\dfrac{2133}{14}\right)\sqrt{3}\\
\end{eqnarray*}