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 \( X=\left(-\dfrac{22}{3}\right)\sqrt{12}+\left(-3\right)\sqrt{12}+\left(-\dfrac{20}{9}\right)\sqrt{12}+\left(-\dfrac{20}{9}\right)\sqrt{12}+\left(\dfrac{59}{3}\right)\sqrt{12}+\left(\left(\dfrac{37}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{12}\right)-\dfrac{1}{6}+\left(\dfrac{71}{5}\right)\sqrt{12}\) et \( Y=-2-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(\left(-8\right)\sqrt{75}\right)-\left(\left(-\dfrac{79}{7}\right)\sqrt{9}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{22}{3}\right)\sqrt{12}+\left(-3\right)\sqrt{12}+\left(-\dfrac{20}{9}\right)\sqrt{12}+\left(-\dfrac{20}{9}\right)\sqrt{12}+\left(\dfrac{59}{3}\right)\sqrt{12}+\left(\left(\dfrac{37}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{12}\right)-\dfrac{1}{6}+\left(\dfrac{71}{5}\right)\sqrt{12}\right)+\left(-2-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(\left(-8\right)\sqrt{75}\right)-\left(\left(-\dfrac{79}{7}\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(-\dfrac{44}{3}\right)\sqrt{3}+\left(-6\right)\sqrt{3}+\left(-\dfrac{40}{9}\right)\sqrt{3}+\left(-\dfrac{40}{9}\right)\sqrt{3}+\left(\dfrac{118}{3}\right)\sqrt{3}+\left(\left(\dfrac{111}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{3}\right)-\dfrac{1}{6}+\left(\dfrac{142}{5}\right)\sqrt{3}\right)+\left(-2+24-\left(\left(\left(-40\right)\sqrt{3}\right)+\dfrac{237}{7}\right)\right)\\
&=&\left(-\dfrac{44}{3}\right)\sqrt{3}+\left(-6\right)\sqrt{3}+\left(-\dfrac{40}{9}\right)\sqrt{3}+\left(-\dfrac{40}{9}\right)\sqrt{3}+\left(\dfrac{118}{3}\right)\sqrt{3}+\left(\left(\dfrac{111}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{3}\right)-\dfrac{1}{6}+\left(\dfrac{142}{5}\right)\sqrt{3}-2+24-\left(\left(\left(-40\right)\sqrt{3}\right)+\dfrac{237}{7}\right)\\
&=&\left(\dfrac{48217}{630}\right)\sqrt{3}-\dfrac{505}{42}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{22}{3}\right)\sqrt{12}+\left(-3\right)\sqrt{12}+\left(-\dfrac{20}{9}\right)\sqrt{12}+\left(-\dfrac{20}{9}\right)\sqrt{12}+\left(\dfrac{59}{3}\right)\sqrt{12}+\left(\left(\dfrac{37}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{12}\right)-\dfrac{1}{6}+\left(\dfrac{71}{5}\right)\sqrt{12}\right)-\left(-2-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(\left(-8\right)\sqrt{75}\right)-\left(\left(-\dfrac{79}{7}\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(-\dfrac{44}{3}\right)\sqrt{3}+\left(-6\right)\sqrt{3}+\left(-\dfrac{40}{9}\right)\sqrt{3}+\left(-\dfrac{40}{9}\right)\sqrt{3}+\left(\dfrac{118}{3}\right)\sqrt{3}+\left(\left(\dfrac{111}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{3}\right)-\dfrac{1}{6}+\left(\dfrac{142}{5}\right)\sqrt{3}\right)-\left(-2+24-\left(\left(\left(-40\right)\sqrt{3}\right)+\dfrac{237}{7}\right)\right)\\
&=&\left(\left(\dfrac{23017}{630}\right)\sqrt{3}-\dfrac{1}{6}\right)-\left(-\dfrac{83}{7}+\left(40\right)\sqrt{3}\right)\\
&=&\left(\dfrac{23017}{630}\right)\sqrt{3}-\dfrac{1}{6}+\dfrac{83}{7}+\left(-40\right)\sqrt{3}\\
&=&\left(-\dfrac{2183}{630}\right)\sqrt{3}+\dfrac{491}{42}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{22}{3}\right)\sqrt{12}+\left(-3\right)\sqrt{12}+\left(-\dfrac{20}{9}\right)\sqrt{12}+\left(-\dfrac{20}{9}\right)\sqrt{12}+\left(\dfrac{59}{3}\right)\sqrt{12}+\left(\left(\dfrac{37}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{12}\right)-\dfrac{1}{6}+\left(\dfrac{71}{5}\right)\sqrt{12}\right)\times\left(-2-\left(\left(-8\right)\sqrt{9}\right)-\left(\left(\left(-8\right)\sqrt{75}\right)-\left(\left(-\dfrac{79}{7}\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(-\dfrac{44}{3}\right)\sqrt{3}+\left(-6\right)\sqrt{3}+\left(-\dfrac{40}{9}\right)\sqrt{3}+\left(-\dfrac{40}{9}\right)\sqrt{3}+\left(\dfrac{118}{3}\right)\sqrt{3}+\left(\left(\dfrac{111}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{3}\right)-\dfrac{1}{6}+\left(\dfrac{142}{5}\right)\sqrt{3}\right)\times\left(-2+24-\left(\left(\left(-40\right)\sqrt{3}\right)+\dfrac{237}{7}\right)\right)\\
&=&\left(\left(\dfrac{23017}{630}\right)\sqrt{3}-\dfrac{1}{6}\right)\left(-\dfrac{83}{7}+\left(40\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{1939811}{4410}\right)\sqrt{3}+\left(\dfrac{92068}{63}\right)\sqrt{9}+\dfrac{83}{42}\\
&=&\left(-\dfrac{1939811}{4410}\right)\sqrt{3}+\dfrac{26317}{6}\\
\end{eqnarray*}