L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{58}{9}+\left(-5\right)\sqrt{27}+\left(-\dfrac{38}{3}\right)\sqrt{12}+\left(-\dfrac{3}{4}\right)\sqrt{9}+\left(\dfrac{70}{3}\right)\sqrt{27}+\left(-\dfrac{52}{9}\right)\sqrt{12}+\left(-\dfrac{77}{5}\right)\sqrt{9}\) et \( Y=\left(-\dfrac{5}{2}\right)\sqrt{27}+\left(\dfrac{59}{2}\right)\sqrt{12}+\left(\dfrac{17}{6}\right)\sqrt{9}+\left(\dfrac{59}{8}\right)\sqrt{12}+\left(-\dfrac{7}{2}\right)\sqrt{12}+\left(\dfrac{14}{3}\right)\sqrt{9}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{58}{9}+\left(-5\right)\sqrt{27}+\left(-\dfrac{38}{3}\right)\sqrt{12}+\left(-\dfrac{3}{4}\right)\sqrt{9}+\left(\dfrac{70}{3}\right)\sqrt{27}+\left(-\dfrac{52}{9}\right)\sqrt{12}+\left(-\dfrac{77}{5}\right)\sqrt{9}\right)+\left(\left(-\dfrac{5}{2}\right)\sqrt{27}+\left(\dfrac{59}{2}\right)\sqrt{12}+\left(\dfrac{17}{6}\right)\sqrt{9}+\left(\dfrac{59}{8}\right)\sqrt{12}+\left(-\dfrac{7}{2}\right)\sqrt{12}+\left(\dfrac{14}{3}\right)\sqrt{9}\right)\\
&=&\left(-\dfrac{58}{9}+\left(-15\right)\sqrt{3}+\left(-\dfrac{76}{3}\right)\sqrt{3}-\dfrac{9}{4}+\left(70\right)\sqrt{3}+\left(-\dfrac{104}{9}\right)\sqrt{3}-\dfrac{231}{5}\right)+\left(\left(-\dfrac{15}{2}\right)\sqrt{3}+\left(59\right)\sqrt{3}+\dfrac{17}{2}+\left(\dfrac{59}{4}\right)\sqrt{3}+\left(-7\right)\sqrt{3}+14\right)\\
&=&-\dfrac{58}{9}+\left(-15\right)\sqrt{3}+\left(-\dfrac{76}{3}\right)\sqrt{3}-\dfrac{9}{4}+\left(70\right)\sqrt{3}+\left(-\dfrac{104}{9}\right)\sqrt{3}-\dfrac{231}{5}+\left(-\dfrac{15}{2}\right)\sqrt{3}+\left(59\right)\sqrt{3}+\dfrac{17}{2}+\left(\dfrac{59}{4}\right)\sqrt{3}+\left(-7\right)\sqrt{3}+14\\
&=&-\dfrac{5831}{180}+\left(\dfrac{2785}{36}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{58}{9}+\left(-5\right)\sqrt{27}+\left(-\dfrac{38}{3}\right)\sqrt{12}+\left(-\dfrac{3}{4}\right)\sqrt{9}+\left(\dfrac{70}{3}\right)\sqrt{27}+\left(-\dfrac{52}{9}\right)\sqrt{12}+\left(-\dfrac{77}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{27}+\left(\dfrac{59}{2}\right)\sqrt{12}+\left(\dfrac{17}{6}\right)\sqrt{9}+\left(\dfrac{59}{8}\right)\sqrt{12}+\left(-\dfrac{7}{2}\right)\sqrt{12}+\left(\dfrac{14}{3}\right)\sqrt{9}\right)\\
&=&\left(-\dfrac{58}{9}+\left(-15\right)\sqrt{3}+\left(-\dfrac{76}{3}\right)\sqrt{3}-\dfrac{9}{4}+\left(70\right)\sqrt{3}+\left(-\dfrac{104}{9}\right)\sqrt{3}-\dfrac{231}{5}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{3}+\left(59\right)\sqrt{3}+\dfrac{17}{2}+\left(\dfrac{59}{4}\right)\sqrt{3}+\left(-7\right)\sqrt{3}+14\right)\\
&=&\left(-\dfrac{9881}{180}+\left(\dfrac{163}{9}\right)\sqrt{3}\right)-\left(\left(\dfrac{237}{4}\right)\sqrt{3}+\dfrac{45}{2}\right)\\
&=&-\dfrac{9881}{180}+\left(\dfrac{163}{9}\right)\sqrt{3}+\left(-\dfrac{237}{4}\right)\sqrt{3}-\dfrac{45}{2}\\
&=&-\dfrac{13931}{180}+\left(-\dfrac{1481}{36}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{58}{9}+\left(-5\right)\sqrt{27}+\left(-\dfrac{38}{3}\right)\sqrt{12}+\left(-\dfrac{3}{4}\right)\sqrt{9}+\left(\dfrac{70}{3}\right)\sqrt{27}+\left(-\dfrac{52}{9}\right)\sqrt{12}+\left(-\dfrac{77}{5}\right)\sqrt{9}\right)\times\left(\left(-\dfrac{5}{2}\right)\sqrt{27}+\left(\dfrac{59}{2}\right)\sqrt{12}+\left(\dfrac{17}{6}\right)\sqrt{9}+\left(\dfrac{59}{8}\right)\sqrt{12}+\left(-\dfrac{7}{2}\right)\sqrt{12}+\left(\dfrac{14}{3}\right)\sqrt{9}\right)\\
&=&\left(-\dfrac{58}{9}+\left(-15\right)\sqrt{3}+\left(-\dfrac{76}{3}\right)\sqrt{3}-\dfrac{9}{4}+\left(70\right)\sqrt{3}+\left(-\dfrac{104}{9}\right)\sqrt{3}-\dfrac{231}{5}\right)\times\left(\left(-\dfrac{15}{2}\right)\sqrt{3}+\left(59\right)\sqrt{3}+\dfrac{17}{2}+\left(\dfrac{59}{4}\right)\sqrt{3}+\left(-7\right)\sqrt{3}+14\right)\\
&=&\left(-\dfrac{9881}{180}+\left(\dfrac{163}{9}\right)\sqrt{3}\right)\left(\left(\dfrac{237}{4}\right)\sqrt{3}+\dfrac{45}{2}\right)\\
&=&\left(-\dfrac{682799}{240}\right)\sqrt{3}-\dfrac{9881}{8}+\left(\dfrac{12877}{12}\right)\sqrt{9}\\
&=&\left(-\dfrac{682799}{240}\right)\sqrt{3}+\dfrac{15873}{8}\\
\end{eqnarray*}