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 \( X=\left(\dfrac{19}{2}\right)\sqrt{12}\) et \( Y=\left(\left(\dfrac{27}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{27}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{27}\right)+\left(\dfrac{70}{9}\right)\sqrt{75}+\left(\dfrac{78}{7}\right)\sqrt{27}\) . 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{19}{2}\right)\sqrt{12}\right)+\left(\left(\left(\dfrac{27}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{27}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{27}\right)+\left(\dfrac{70}{9}\right)\sqrt{75}+\left(\dfrac{78}{7}\right)\sqrt{27}\right)\\
&=&\left(\left(19\right)\sqrt{3}\right)+\left(\left(\left(\dfrac{81}{4}\right)\sqrt{3}\right)-\left(\left(-3\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{189}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{231}{4}\right)\sqrt{3}\right)+\left(\dfrac{350}{9}\right)\sqrt{3}+\left(\dfrac{234}{7}\right)\sqrt{3}\right)\\
&=&\left(19\right)\sqrt{3}+\left(\left(\dfrac{81}{4}\right)\sqrt{3}\right)-\left(\left(-3\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{189}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{231}{4}\right)\sqrt{3}\right)+\left(\dfrac{350}{9}\right)\sqrt{3}+\left(\dfrac{234}{7}\right)\sqrt{3}\\
&=&\left(\dfrac{60577}{504}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{19}{2}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{27}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{27}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{27}\right)+\left(\dfrac{70}{9}\right)\sqrt{75}+\left(\dfrac{78}{7}\right)\sqrt{27}\right)\\
&=&\left(\left(19\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{81}{4}\right)\sqrt{3}\right)-\left(\left(-3\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{189}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{231}{4}\right)\sqrt{3}\right)+\left(\dfrac{350}{9}\right)\sqrt{3}+\left(\dfrac{234}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(19\right)\sqrt{3}\right)-\left(\left(\dfrac{51001}{504}\right)\sqrt{3}\right)\\
&=&\left(19\right)\sqrt{3}+\left(-\dfrac{51001}{504}\right)\sqrt{3}\\
&=&\left(-\dfrac{41425}{504}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{19}{2}\right)\sqrt{12}\right)\times\left(\left(\left(\dfrac{27}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{27}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{27}\right)+\left(\dfrac{70}{9}\right)\sqrt{75}+\left(\dfrac{78}{7}\right)\sqrt{27}\right)\\
&=&\left(\left(19\right)\sqrt{3}\right)\times\left(\left(\left(\dfrac{81}{4}\right)\sqrt{3}\right)-\left(\left(-3\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{189}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{231}{4}\right)\sqrt{3}\right)+\left(\dfrac{350}{9}\right)\sqrt{3}+\left(\dfrac{234}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(19\right)\sqrt{3}\right)\left(\left(\dfrac{51001}{504}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{969019}{504}\right)\sqrt{9}\\
&=&\dfrac{969019}{168}\\
\end{eqnarray*}