L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-5-\left(\left(-6\right)\sqrt{75}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{12}\right)+\left(\left(-3\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}\right)+\left(\dfrac{17}{5}\right)\sqrt{9}\) et \( Y=\left(\dfrac{13}{8}\right)\sqrt{12}+\left(-\dfrac{10}{7}\right)\sqrt{75}+\left(-7\right)\sqrt{27}+\left(-\dfrac{17}{3}\right)\sqrt{27}+\dfrac{73}{9}-\left(\left(0\right)\sqrt{27}\right)-\left(\left(\dfrac{66}{7}\right)\sqrt{27}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-5-\left(\left(-6\right)\sqrt{75}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{12}\right)+\left(\left(-3\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}\right)+\left(\dfrac{17}{5}\right)\sqrt{9}\right)+\left(\left(\dfrac{13}{8}\right)\sqrt{12}+\left(-\dfrac{10}{7}\right)\sqrt{75}+\left(-7\right)\sqrt{27}+\left(-\dfrac{17}{3}\right)\sqrt{27}+\dfrac{73}{9}-\left(\left(0\right)\sqrt{27}\right)-\left(\left(\dfrac{66}{7}\right)\sqrt{27}\right)\right)\\
&=&\left(-5-\left(\left(-30\right)\sqrt{3}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{3}\right)+\left(\left(-9\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)+\dfrac{51}{5}\right)+\left(\left(\dfrac{13}{4}\right)\sqrt{3}+\left(-\dfrac{50}{7}\right)\sqrt{3}+\left(-21\right)\sqrt{3}+\left(-17\right)\sqrt{3}+\dfrac{73}{9}-\left(\left(0\right)\sqrt{3}\right)-\left(\left(\dfrac{198}{7}\right)\sqrt{3}\right)\right)\\
&=&-5-\left(\left(-30\right)\sqrt{3}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{3}\right)+\left(\left(-9\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)+\dfrac{51}{5}+\left(\dfrac{13}{4}\right)\sqrt{3}+\left(-\dfrac{50}{7}\right)\sqrt{3}+\left(-21\right)\sqrt{3}+\left(-17\right)\sqrt{3}+\dfrac{73}{9}-\left(\left(0\right)\sqrt{3}\right)-\left(\left(\dfrac{198}{7}\right)\sqrt{3}\right)\\
&=&\dfrac{599}{45}+\left(-\dfrac{16369}{252}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-5-\left(\left(-6\right)\sqrt{75}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{12}\right)+\left(\left(-3\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}\right)+\left(\dfrac{17}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{13}{8}\right)\sqrt{12}+\left(-\dfrac{10}{7}\right)\sqrt{75}+\left(-7\right)\sqrt{27}+\left(-\dfrac{17}{3}\right)\sqrt{27}+\dfrac{73}{9}-\left(\left(0\right)\sqrt{27}\right)-\left(\left(\dfrac{66}{7}\right)\sqrt{27}\right)\right)\\
&=&\left(-5-\left(\left(-30\right)\sqrt{3}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{3}\right)+\left(\left(-9\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)+\dfrac{51}{5}\right)-\left(\left(\dfrac{13}{4}\right)\sqrt{3}+\left(-\dfrac{50}{7}\right)\sqrt{3}+\left(-21\right)\sqrt{3}+\left(-17\right)\sqrt{3}+\dfrac{73}{9}-\left(\left(0\right)\sqrt{3}\right)-\left(\left(\dfrac{198}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{26}{5}+\left(\dfrac{47}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{1965}{28}\right)\sqrt{3}+\dfrac{73}{9}\right)\\
&=&\dfrac{26}{5}+\left(\dfrac{47}{9}\right)\sqrt{3}+\left(\dfrac{1965}{28}\right)\sqrt{3}-\dfrac{73}{9}\\
&=&-\dfrac{131}{45}+\left(\dfrac{19001}{252}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-5-\left(\left(-6\right)\sqrt{75}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{12}\right)+\left(\left(-3\right)\sqrt{27}\right)-\left(\left(2\right)\sqrt{75}\right)+\left(\dfrac{17}{5}\right)\sqrt{9}\right)\times\left(\left(\dfrac{13}{8}\right)\sqrt{12}+\left(-\dfrac{10}{7}\right)\sqrt{75}+\left(-7\right)\sqrt{27}+\left(-\dfrac{17}{3}\right)\sqrt{27}+\dfrac{73}{9}-\left(\left(0\right)\sqrt{27}\right)-\left(\left(\dfrac{66}{7}\right)\sqrt{27}\right)\right)\\
&=&\left(-5-\left(\left(-30\right)\sqrt{3}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{3}\right)+\left(\left(-9\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)+\dfrac{51}{5}\right)\times\left(\left(\dfrac{13}{4}\right)\sqrt{3}+\left(-\dfrac{50}{7}\right)\sqrt{3}+\left(-21\right)\sqrt{3}+\left(-17\right)\sqrt{3}+\dfrac{73}{9}-\left(\left(0\right)\sqrt{3}\right)-\left(\left(\dfrac{198}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{26}{5}+\left(\dfrac{47}{9}\right)\sqrt{3}\right)\left(\left(-\dfrac{1965}{28}\right)\sqrt{3}+\dfrac{73}{9}\right)\\
&=&\left(-\dfrac{365795}{1134}\right)\sqrt{3}+\dfrac{1898}{45}+\left(-\dfrac{30785}{84}\right)\sqrt{9}\\
&=&\left(-\dfrac{365795}{1134}\right)\sqrt{3}-\dfrac{1332181}{1260}\\
\end{eqnarray*}