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 \( X=\left(-6\right)\sqrt{75}\) et \( Y=\left(\dfrac{24}{7}\right)\sqrt{9}+\left(-\dfrac{80}{9}\right)\sqrt{9}+\left(\dfrac{5}{3}\right)\sqrt{12}+\left(-\dfrac{43}{5}\right)\sqrt{75}+\left(\dfrac{9}{7}\right)\sqrt{12}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-6\right)\sqrt{75}\right)+\left(\left(\dfrac{24}{7}\right)\sqrt{9}+\left(-\dfrac{80}{9}\right)\sqrt{9}+\left(\dfrac{5}{3}\right)\sqrt{12}+\left(-\dfrac{43}{5}\right)\sqrt{75}+\left(\dfrac{9}{7}\right)\sqrt{12}\right)\\
&=&\left(\left(-30\right)\sqrt{3}\right)+\left(\dfrac{72}{7}-\dfrac{80}{3}+\left(\dfrac{10}{3}\right)\sqrt{3}+\left(-43\right)\sqrt{3}+\left(\dfrac{18}{7}\right)\sqrt{3}\right)\\
&=&\left(-30\right)\sqrt{3}+\dfrac{72}{7}-\dfrac{80}{3}+\left(\dfrac{10}{3}\right)\sqrt{3}+\left(-43\right)\sqrt{3}+\left(\dfrac{18}{7}\right)\sqrt{3}\\
&=&\left(-\dfrac{1409}{21}\right)\sqrt{3}-\dfrac{344}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-6\right)\sqrt{75}\right)-\left(\left(\dfrac{24}{7}\right)\sqrt{9}+\left(-\dfrac{80}{9}\right)\sqrt{9}+\left(\dfrac{5}{3}\right)\sqrt{12}+\left(-\dfrac{43}{5}\right)\sqrt{75}+\left(\dfrac{9}{7}\right)\sqrt{12}\right)\\
&=&\left(\left(-30\right)\sqrt{3}\right)-\left(\dfrac{72}{7}-\dfrac{80}{3}+\left(\dfrac{10}{3}\right)\sqrt{3}+\left(-43\right)\sqrt{3}+\left(\dfrac{18}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(-30\right)\sqrt{3}\right)-\left(-\dfrac{344}{21}+\left(-\dfrac{779}{21}\right)\sqrt{3}\right)\\
&=&\left(-30\right)\sqrt{3}+\dfrac{344}{21}+\left(\dfrac{779}{21}\right)\sqrt{3}\\
&=&\left(\dfrac{149}{21}\right)\sqrt{3}+\dfrac{344}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-6\right)\sqrt{75}\right)\times\left(\left(\dfrac{24}{7}\right)\sqrt{9}+\left(-\dfrac{80}{9}\right)\sqrt{9}+\left(\dfrac{5}{3}\right)\sqrt{12}+\left(-\dfrac{43}{5}\right)\sqrt{75}+\left(\dfrac{9}{7}\right)\sqrt{12}\right)\\
&=&\left(\left(-30\right)\sqrt{3}\right)\times\left(\dfrac{72}{7}-\dfrac{80}{3}+\left(\dfrac{10}{3}\right)\sqrt{3}+\left(-43\right)\sqrt{3}+\left(\dfrac{18}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(-30\right)\sqrt{3}\right)\left(-\dfrac{344}{21}+\left(-\dfrac{779}{21}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{3440}{7}\right)\sqrt{3}+\left(\dfrac{7790}{7}\right)\sqrt{9}\\
&=&\left(\dfrac{3440}{7}\right)\sqrt{3}+\dfrac{23370}{7}\\
\end{eqnarray*}