L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{6}{5}\right)\sqrt{27}\right)-\left(\left(-6\right)\sqrt{9}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{12}\right)\) et \( Y=\left(\dfrac{79}{2}\right)\sqrt{75}+\left(\dfrac{63}{5}\right)\sqrt{12}+\left(\left(-\dfrac{67}{4}\right)\sqrt{75}\right)-\left(\left(-4\right)\sqrt{9}\right)+\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{27}\right)\) . 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{6}{5}\right)\sqrt{27}\right)-\left(\left(-6\right)\sqrt{9}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{12}\right)\right)+\left(\left(\dfrac{79}{2}\right)\sqrt{75}+\left(\dfrac{63}{5}\right)\sqrt{12}+\left(\left(-\dfrac{67}{4}\right)\sqrt{75}\right)-\left(\left(-4\right)\sqrt{9}\right)+\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(\dfrac{18}{5}\right)\sqrt{3}\right)+18-\left(\left(\dfrac{11}{2}\right)\sqrt{3}\right)\right)+\left(\left(\dfrac{395}{2}\right)\sqrt{3}+\left(\dfrac{126}{5}\right)\sqrt{3}+\left(\left(-\dfrac{335}{4}\right)\sqrt{3}\right)+12+\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\dfrac{18}{5}\right)\sqrt{3}\right)+18-\left(\left(\dfrac{11}{2}\right)\sqrt{3}\right)+\left(\dfrac{395}{2}\right)\sqrt{3}+\left(\dfrac{126}{5}\right)\sqrt{3}+\left(\left(-\dfrac{335}{4}\right)\sqrt{3}\right)+12+\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}\right)\\
&=&\left(\dfrac{2011}{20}\right)\sqrt{3}+30\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{6}{5}\right)\sqrt{27}\right)-\left(\left(-6\right)\sqrt{9}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{12}\right)\right)-\left(\left(\dfrac{79}{2}\right)\sqrt{75}+\left(\dfrac{63}{5}\right)\sqrt{12}+\left(\left(-\dfrac{67}{4}\right)\sqrt{75}\right)-\left(\left(-4\right)\sqrt{9}\right)+\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(\dfrac{18}{5}\right)\sqrt{3}\right)+18-\left(\left(\dfrac{11}{2}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{395}{2}\right)\sqrt{3}+\left(\dfrac{126}{5}\right)\sqrt{3}+\left(\left(-\dfrac{335}{4}\right)\sqrt{3}\right)+12+\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{19}{10}\right)\sqrt{3}+18\right)-\left(\left(\dfrac{2049}{20}\right)\sqrt{3}+12\right)\\
&=&\left(-\dfrac{19}{10}\right)\sqrt{3}+18+\left(-\dfrac{2049}{20}\right)\sqrt{3}-12\\
&=&\left(-\dfrac{2087}{20}\right)\sqrt{3}+6\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{6}{5}\right)\sqrt{27}\right)-\left(\left(-6\right)\sqrt{9}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{12}\right)\right)\times\left(\left(\dfrac{79}{2}\right)\sqrt{75}+\left(\dfrac{63}{5}\right)\sqrt{12}+\left(\left(-\dfrac{67}{4}\right)\sqrt{75}\right)-\left(\left(-4\right)\sqrt{9}\right)+\left(\left(-\dfrac{15}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(\dfrac{18}{5}\right)\sqrt{3}\right)+18-\left(\left(\dfrac{11}{2}\right)\sqrt{3}\right)\right)\times\left(\left(\dfrac{395}{2}\right)\sqrt{3}+\left(\dfrac{126}{5}\right)\sqrt{3}+\left(\left(-\dfrac{335}{4}\right)\sqrt{3}\right)+12+\left(\left(-\dfrac{45}{2}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{19}{10}\right)\sqrt{3}+18\right)\left(\left(\dfrac{2049}{20}\right)\sqrt{3}+12\right)\\
&=&\left(-\dfrac{38931}{200}\right)\sqrt{9}+\left(\dfrac{18213}{10}\right)\sqrt{3}+216\\
&=&-\dfrac{73593}{200}+\left(\dfrac{18213}{10}\right)\sqrt{3}\\
\end{eqnarray*}