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 \( X=\left(\dfrac{11}{3}\right)\sqrt{9}+\left(\dfrac{20}{3}\right)\sqrt{27}+\left(\dfrac{19}{3}\right)\sqrt{27}+\left(\dfrac{54}{5}\right)\sqrt{75}+\left(\dfrac{16}{3}\right)\sqrt{75}+\left(-\dfrac{50}{9}\right)\sqrt{12}+\left(\dfrac{79}{8}\right)\sqrt{27}+\left(\dfrac{13}{2}\right)\sqrt{27}+\dfrac{24}{7}\) et \( Y=\left(7\right)\sqrt{27}\) . 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{11}{3}\right)\sqrt{9}+\left(\dfrac{20}{3}\right)\sqrt{27}+\left(\dfrac{19}{3}\right)\sqrt{27}+\left(\dfrac{54}{5}\right)\sqrt{75}+\left(\dfrac{16}{3}\right)\sqrt{75}+\left(-\dfrac{50}{9}\right)\sqrt{12}+\left(\dfrac{79}{8}\right)\sqrt{27}+\left(\dfrac{13}{2}\right)\sqrt{27}+\dfrac{24}{7}\right)+\left(\left(7\right)\sqrt{27}\right)\\
&=&\left(11+\left(20\right)\sqrt{3}+\left(19\right)\sqrt{3}+\left(54\right)\sqrt{3}+\left(\dfrac{80}{3}\right)\sqrt{3}+\left(-\dfrac{100}{9}\right)\sqrt{3}+\left(\dfrac{237}{8}\right)\sqrt{3}+\left(\dfrac{39}{2}\right)\sqrt{3}+\dfrac{24}{7}\right)+\left(\left(21\right)\sqrt{3}\right)\\
&=&11+\left(20\right)\sqrt{3}+\left(19\right)\sqrt{3}+\left(54\right)\sqrt{3}+\left(\dfrac{80}{3}\right)\sqrt{3}+\left(-\dfrac{100}{9}\right)\sqrt{3}+\left(\dfrac{237}{8}\right)\sqrt{3}+\left(\dfrac{39}{2}\right)\sqrt{3}+\dfrac{24}{7}+\left(21\right)\sqrt{3}\\
&=&\dfrac{101}{7}+\left(\dfrac{12865}{72}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{11}{3}\right)\sqrt{9}+\left(\dfrac{20}{3}\right)\sqrt{27}+\left(\dfrac{19}{3}\right)\sqrt{27}+\left(\dfrac{54}{5}\right)\sqrt{75}+\left(\dfrac{16}{3}\right)\sqrt{75}+\left(-\dfrac{50}{9}\right)\sqrt{12}+\left(\dfrac{79}{8}\right)\sqrt{27}+\left(\dfrac{13}{2}\right)\sqrt{27}+\dfrac{24}{7}\right)-\left(\left(7\right)\sqrt{27}\right)\\
&=&\left(11+\left(20\right)\sqrt{3}+\left(19\right)\sqrt{3}+\left(54\right)\sqrt{3}+\left(\dfrac{80}{3}\right)\sqrt{3}+\left(-\dfrac{100}{9}\right)\sqrt{3}+\left(\dfrac{237}{8}\right)\sqrt{3}+\left(\dfrac{39}{2}\right)\sqrt{3}+\dfrac{24}{7}\right)-\left(\left(21\right)\sqrt{3}\right)\\
&=&\left(\dfrac{101}{7}+\left(\dfrac{11353}{72}\right)\sqrt{3}\right)-\left(\left(21\right)\sqrt{3}\right)\\
&=&\dfrac{101}{7}+\left(\dfrac{11353}{72}\right)\sqrt{3}+\left(-21\right)\sqrt{3}\\
&=&\dfrac{101}{7}+\left(\dfrac{9841}{72}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{11}{3}\right)\sqrt{9}+\left(\dfrac{20}{3}\right)\sqrt{27}+\left(\dfrac{19}{3}\right)\sqrt{27}+\left(\dfrac{54}{5}\right)\sqrt{75}+\left(\dfrac{16}{3}\right)\sqrt{75}+\left(-\dfrac{50}{9}\right)\sqrt{12}+\left(\dfrac{79}{8}\right)\sqrt{27}+\left(\dfrac{13}{2}\right)\sqrt{27}+\dfrac{24}{7}\right)\times\left(\left(7\right)\sqrt{27}\right)\\
&=&\left(11+\left(20\right)\sqrt{3}+\left(19\right)\sqrt{3}+\left(54\right)\sqrt{3}+\left(\dfrac{80}{3}\right)\sqrt{3}+\left(-\dfrac{100}{9}\right)\sqrt{3}+\left(\dfrac{237}{8}\right)\sqrt{3}+\left(\dfrac{39}{2}\right)\sqrt{3}+\dfrac{24}{7}\right)\times\left(\left(21\right)\sqrt{3}\right)\\
&=&\left(\dfrac{101}{7}+\left(\dfrac{11353}{72}\right)\sqrt{3}\right)\left(\left(21\right)\sqrt{3}\right)\\
&=&\left(303\right)\sqrt{3}+\left(\dfrac{79471}{24}\right)\sqrt{9}\\
&=&\left(303\right)\sqrt{3}+\dfrac{79471}{8}\\
\end{eqnarray*}