L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{23}{8}\right)\sqrt{75}\) et \( Y=-\dfrac{14}{3}+\left(\left(-\dfrac{26}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{9}\right)+\left(\left(-\dfrac{18}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{47}{3}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(7\right)\sqrt{75}\right)+\left(\dfrac{1}{2}\right)\sqrt{27}+\dfrac{67}{2}-3+\left(-\dfrac{35}{2}\right)\sqrt{75}\) . 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{23}{8}\right)\sqrt{75}\right)+\left(-\dfrac{14}{3}+\left(\left(-\dfrac{26}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{9}\right)+\left(\left(-\dfrac{18}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{47}{3}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(7\right)\sqrt{75}\right)+\left(\dfrac{1}{2}\right)\sqrt{27}+\dfrac{67}{2}-3+\left(-\dfrac{35}{2}\right)\sqrt{75}\right)\\
&=&\left(\left(\dfrac{115}{8}\right)\sqrt{3}\right)+\left(-\dfrac{14}{3}+\left(\left(-26\right)\sqrt{3}\right)+\dfrac{65}{3}-\dfrac{54}{5}-\left(\left(-\dfrac{94}{3}\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)+\left(\dfrac{3}{2}\right)\sqrt{3}+\dfrac{67}{2}-3+\left(-\dfrac{175}{2}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{115}{8}\right)\sqrt{3}-\dfrac{14}{3}+\left(\left(-26\right)\sqrt{3}\right)+\dfrac{65}{3}-\dfrac{54}{5}-\left(\left(-\dfrac{94}{3}\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)+\left(\dfrac{3}{2}\right)\sqrt{3}+\dfrac{67}{2}-3+\left(-\dfrac{175}{2}\right)\sqrt{3}\\
&=&\left(-\dfrac{2191}{24}\right)\sqrt{3}+\dfrac{367}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{23}{8}\right)\sqrt{75}\right)-\left(-\dfrac{14}{3}+\left(\left(-\dfrac{26}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{9}\right)+\left(\left(-\dfrac{18}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{47}{3}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(7\right)\sqrt{75}\right)+\left(\dfrac{1}{2}\right)\sqrt{27}+\dfrac{67}{2}-3+\left(-\dfrac{35}{2}\right)\sqrt{75}\right)\\
&=&\left(\left(\dfrac{115}{8}\right)\sqrt{3}\right)-\left(-\dfrac{14}{3}+\left(\left(-26\right)\sqrt{3}\right)+\dfrac{65}{3}-\dfrac{54}{5}-\left(\left(-\dfrac{94}{3}\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)+\left(\dfrac{3}{2}\right)\sqrt{3}+\dfrac{67}{2}-3+\left(-\dfrac{175}{2}\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{115}{8}\right)\sqrt{3}\right)-\left(\dfrac{367}{10}+\left(-\dfrac{317}{3}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{115}{8}\right)\sqrt{3}+-\dfrac{367}{10}+\left(\dfrac{317}{3}\right)\sqrt{3}\\
&=&\left(\dfrac{2881}{24}\right)\sqrt{3}-\dfrac{367}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{23}{8}\right)\sqrt{75}\right)\times\left(-\dfrac{14}{3}+\left(\left(-\dfrac{26}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{9}\right)+\left(\left(-\dfrac{18}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{47}{3}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(7\right)\sqrt{75}\right)+\left(\dfrac{1}{2}\right)\sqrt{27}+\dfrac{67}{2}-3+\left(-\dfrac{35}{2}\right)\sqrt{75}\right)\\
&=&\left(\left(\dfrac{115}{8}\right)\sqrt{3}\right)\times\left(-\dfrac{14}{3}+\left(\left(-26\right)\sqrt{3}\right)+\dfrac{65}{3}-\dfrac{54}{5}-\left(\left(-\dfrac{94}{3}\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)+\left(\dfrac{3}{2}\right)\sqrt{3}+\dfrac{67}{2}-3+\left(-\dfrac{175}{2}\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{115}{8}\right)\sqrt{3}\right)\left(\dfrac{367}{10}+\left(-\dfrac{317}{3}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{8441}{16}\right)\sqrt{3}+\left(-\dfrac{36455}{24}\right)\sqrt{9}\\
&=&\left(\dfrac{8441}{16}\right)\sqrt{3}-\dfrac{36455}{8}\\
\end{eqnarray*}