L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{31}{3}+\dfrac{79}{9}\right)-\left(\left(-\dfrac{17}{8}\right)\sqrt{27}\right)-\left(\left(7\right)\sqrt{12}-1+\left(-\dfrac{17}{4}\right)\sqrt{27}\right)-\left(\left(\left(-\dfrac{29}{7}\right)\sqrt{9}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{27}\right)\right)\) et \( Y=\left(-\dfrac{47}{5}\right)\sqrt{75}\) . 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{31}{3}+\dfrac{79}{9}\right)-\left(\left(-\dfrac{17}{8}\right)\sqrt{27}\right)-\left(\left(7\right)\sqrt{12}-1+\left(-\dfrac{17}{4}\right)\sqrt{27}\right)-\left(\left(\left(-\dfrac{29}{7}\right)\sqrt{9}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{27}\right)\right)\right)+\left(\left(-\dfrac{47}{5}\right)\sqrt{75}\right)\\
&=&\left(\left(\dfrac{31}{3}+\dfrac{79}{9}\right)-\left(\left(-\dfrac{51}{8}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}-1+\left(-\dfrac{51}{4}\right)\sqrt{3}\right)-\left(-\dfrac{87}{7}-\left(\left(\dfrac{65}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{270}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{225}{2}\right)\sqrt{3}\right)\right)\right)+\left(\left(-47\right)\sqrt{3}\right)\\
&=&\left(\dfrac{31}{3}+\dfrac{79}{9}\right)-\left(\left(-\dfrac{51}{8}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}-1+\left(-\dfrac{51}{4}\right)\sqrt{3}\right)-\left(-\dfrac{87}{7}-\left(\left(\dfrac{65}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{270}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{225}{2}\right)\sqrt{3}\right)\right)+\left(-47\right)\sqrt{3}\\
&=&\dfrac{2050}{63}+\left(-\dfrac{15815}{168}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{31}{3}+\dfrac{79}{9}\right)-\left(\left(-\dfrac{17}{8}\right)\sqrt{27}\right)-\left(\left(7\right)\sqrt{12}-1+\left(-\dfrac{17}{4}\right)\sqrt{27}\right)-\left(\left(\left(-\dfrac{29}{7}\right)\sqrt{9}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{27}\right)\right)\right)-\left(\left(-\dfrac{47}{5}\right)\sqrt{75}\right)\\
&=&\left(\left(\dfrac{31}{3}+\dfrac{79}{9}\right)-\left(\left(-\dfrac{51}{8}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}-1+\left(-\dfrac{51}{4}\right)\sqrt{3}\right)-\left(-\dfrac{87}{7}-\left(\left(\dfrac{65}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{270}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{225}{2}\right)\sqrt{3}\right)\right)\right)-\left(\left(-47\right)\sqrt{3}\right)\\
&=&\left(\dfrac{2050}{63}+\left(-\dfrac{7919}{168}\right)\sqrt{3}\right)-\left(\left(-47\right)\sqrt{3}\right)\\
&=&\dfrac{2050}{63}+\left(-\dfrac{7919}{168}\right)\sqrt{3}+\left(47\right)\sqrt{3}\\
&=&\dfrac{2050}{63}+\left(-\dfrac{23}{168}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{31}{3}+\dfrac{79}{9}\right)-\left(\left(-\dfrac{17}{8}\right)\sqrt{27}\right)-\left(\left(7\right)\sqrt{12}-1+\left(-\dfrac{17}{4}\right)\sqrt{27}\right)-\left(\left(\left(-\dfrac{29}{7}\right)\sqrt{9}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{27}\right)\right)\right)\times\left(\left(-\dfrac{47}{5}\right)\sqrt{75}\right)\\
&=&\left(\left(\dfrac{31}{3}+\dfrac{79}{9}\right)-\left(\left(-\dfrac{51}{8}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}-1+\left(-\dfrac{51}{4}\right)\sqrt{3}\right)-\left(-\dfrac{87}{7}-\left(\left(\dfrac{65}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{270}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{225}{2}\right)\sqrt{3}\right)\right)\right)\times\left(\left(-47\right)\sqrt{3}\right)\\
&=&\left(\dfrac{2050}{63}+\left(-\dfrac{7919}{168}\right)\sqrt{3}\right)\left(\left(-47\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{96350}{63}\right)\sqrt{3}+\left(\dfrac{372193}{168}\right)\sqrt{9}\\
&=&\left(-\dfrac{96350}{63}\right)\sqrt{3}+\dfrac{372193}{56}\\
\end{eqnarray*}