L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-1\right)\sqrt{75}+\left(\left(-\dfrac{13}{6}\right)\sqrt{27}\right)-\left(\left(\dfrac{47}{4}\right)\sqrt{27}\right)-\dfrac{8}{9}\) et \( Y=\left(\left(\left(-\dfrac{66}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{66}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{20}{9}\right)\sqrt{75}\right)\right)-\dfrac{18}{7}-\left(\left(7\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{75}-4\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-1\right)\sqrt{75}+\left(\left(-\dfrac{13}{6}\right)\sqrt{27}\right)-\left(\left(\dfrac{47}{4}\right)\sqrt{27}\right)-\dfrac{8}{9}\right)+\left(\left(\left(\left(-\dfrac{66}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{66}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{20}{9}\right)\sqrt{75}\right)\right)-\dfrac{18}{7}-\left(\left(7\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{75}-4\right)\right)\\
&=&\left(\left(-5\right)\sqrt{3}+\left(\left(-\dfrac{13}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{141}{4}\right)\sqrt{3}\right)-\dfrac{8}{9}\right)+\left(\left(\left(\left(-\dfrac{132}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{175}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{132}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{100}{9}\right)\sqrt{3}\right)\right)-\dfrac{18}{7}-\left(\left(21\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}-4\right)\right)\\
&=&\left(-5\right)\sqrt{3}+\left(\left(-\dfrac{13}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{141}{4}\right)\sqrt{3}\right)-\dfrac{8}{9}+\left(\left(\left(-\dfrac{132}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{175}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{132}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{100}{9}\right)\sqrt{3}\right)\right)-\dfrac{18}{7}-\left(\left(21\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}-4\right)\\
&=&\left(-\dfrac{2207}{18}\right)\sqrt{3}+\dfrac{34}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-1\right)\sqrt{75}+\left(\left(-\dfrac{13}{6}\right)\sqrt{27}\right)-\left(\left(\dfrac{47}{4}\right)\sqrt{27}\right)-\dfrac{8}{9}\right)-\left(\left(\left(\left(-\dfrac{66}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{66}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{20}{9}\right)\sqrt{75}\right)\right)-\dfrac{18}{7}-\left(\left(7\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{75}-4\right)\right)\\
&=&\left(\left(-5\right)\sqrt{3}+\left(\left(-\dfrac{13}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{141}{4}\right)\sqrt{3}\right)-\dfrac{8}{9}\right)-\left(\left(\left(\left(-\dfrac{132}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{175}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{132}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{100}{9}\right)\sqrt{3}\right)\right)-\dfrac{18}{7}-\left(\left(21\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}-4\right)\right)\\
&=&\left(\left(-\dfrac{187}{4}\right)\sqrt{3}-\dfrac{8}{9}\right)-\left(\left(-\dfrac{2731}{36}\right)\sqrt{3}+\dfrac{10}{7}\right)\\
&=&\left(-\dfrac{187}{4}\right)\sqrt{3}-\dfrac{8}{9}+\left(\dfrac{2731}{36}\right)\sqrt{3}-\dfrac{10}{7}\\
&=&\left(\dfrac{262}{9}\right)\sqrt{3}-\dfrac{146}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-1\right)\sqrt{75}+\left(\left(-\dfrac{13}{6}\right)\sqrt{27}\right)-\left(\left(\dfrac{47}{4}\right)\sqrt{27}\right)-\dfrac{8}{9}\right)\times\left(\left(\left(\left(-\dfrac{66}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{66}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{20}{9}\right)\sqrt{75}\right)\right)-\dfrac{18}{7}-\left(\left(7\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{75}-4\right)\right)\\
&=&\left(\left(-5\right)\sqrt{3}+\left(\left(-\dfrac{13}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{141}{4}\right)\sqrt{3}\right)-\dfrac{8}{9}\right)\times\left(\left(\left(\left(-\dfrac{132}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{175}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{132}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{100}{9}\right)\sqrt{3}\right)\right)-\dfrac{18}{7}-\left(\left(21\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}-4\right)\right)\\
&=&\left(\left(-\dfrac{187}{4}\right)\sqrt{3}-\dfrac{8}{9}\right)\left(\left(-\dfrac{2731}{36}\right)\sqrt{3}+\dfrac{10}{7}\right)\\
&=&\left(\dfrac{510697}{144}\right)\sqrt{9}+\left(\dfrac{733}{1134}\right)\sqrt{3}-\dfrac{80}{63}\\
&=&\dfrac{10723357}{1008}+\left(\dfrac{733}{1134}\right)\sqrt{3}\\
\end{eqnarray*}