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 \( X=\left(-\dfrac{41}{9}\right)\sqrt{12}+4+\left(\dfrac{3}{5}\right)\sqrt{12}+\left(\dfrac{53}{4}\right)\sqrt{27}+\left(-6\right)\sqrt{9}+\left(-6\right)\sqrt{9}\) et \( Y=\left(-\dfrac{2}{9}\right)\sqrt{27}\) . 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{41}{9}\right)\sqrt{12}+4+\left(\dfrac{3}{5}\right)\sqrt{12}+\left(\dfrac{53}{4}\right)\sqrt{27}+\left(-6\right)\sqrt{9}+\left(-6\right)\sqrt{9}\right)+\left(\left(-\dfrac{2}{9}\right)\sqrt{27}\right)\\
&=&\left(\left(-\dfrac{82}{9}\right)\sqrt{3}+4+\left(\dfrac{6}{5}\right)\sqrt{3}+\left(\dfrac{159}{4}\right)\sqrt{3}-18-18\right)+\left(\left(-\dfrac{2}{3}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{82}{9}\right)\sqrt{3}+4+\left(\dfrac{6}{5}\right)\sqrt{3}+\left(\dfrac{159}{4}\right)\sqrt{3}-18-18+\left(-\dfrac{2}{3}\right)\sqrt{3}\\
&=&\left(\dfrac{5611}{180}\right)\sqrt{3}-32\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{41}{9}\right)\sqrt{12}+4+\left(\dfrac{3}{5}\right)\sqrt{12}+\left(\dfrac{53}{4}\right)\sqrt{27}+\left(-6\right)\sqrt{9}+\left(-6\right)\sqrt{9}\right)-\left(\left(-\dfrac{2}{9}\right)\sqrt{27}\right)\\
&=&\left(\left(-\dfrac{82}{9}\right)\sqrt{3}+4+\left(\dfrac{6}{5}\right)\sqrt{3}+\left(\dfrac{159}{4}\right)\sqrt{3}-18-18\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{5731}{180}\right)\sqrt{3}-32\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{5731}{180}\right)\sqrt{3}-32+\left(\dfrac{2}{3}\right)\sqrt{3}\\
&=&\left(\dfrac{5851}{180}\right)\sqrt{3}-32\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{41}{9}\right)\sqrt{12}+4+\left(\dfrac{3}{5}\right)\sqrt{12}+\left(\dfrac{53}{4}\right)\sqrt{27}+\left(-6\right)\sqrt{9}+\left(-6\right)\sqrt{9}\right)\times\left(\left(-\dfrac{2}{9}\right)\sqrt{27}\right)\\
&=&\left(\left(-\dfrac{82}{9}\right)\sqrt{3}+4+\left(\dfrac{6}{5}\right)\sqrt{3}+\left(\dfrac{159}{4}\right)\sqrt{3}-18-18\right)\times\left(\left(-\dfrac{2}{3}\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{5731}{180}\right)\sqrt{3}-32\right)\left(\left(-\dfrac{2}{3}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{5731}{270}\right)\sqrt{9}+\left(\dfrac{64}{3}\right)\sqrt{3}\\
&=&-\dfrac{5731}{90}+\left(\dfrac{64}{3}\right)\sqrt{3}\\
\end{eqnarray*}