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 \( X=\left(-1\right)\sqrt{75}\) et \( Y=\left(\dfrac{75}{7}\right)\sqrt{27}+\left(\dfrac{65}{7}\right)\sqrt{9}+\left(\dfrac{57}{5}\right)\sqrt{27}+\left(4\right)\sqrt{27}+\left(-\dfrac{26}{3}\right)\sqrt{27}+\left(-\dfrac{81}{4}\right)\sqrt{12}+4+\left(0\right)\sqrt{9}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-1\right)\sqrt{75}\right)+\left(\left(\dfrac{75}{7}\right)\sqrt{27}+\left(\dfrac{65}{7}\right)\sqrt{9}+\left(\dfrac{57}{5}\right)\sqrt{27}+\left(4\right)\sqrt{27}+\left(-\dfrac{26}{3}\right)\sqrt{27}+\left(-\dfrac{81}{4}\right)\sqrt{12}+4+\left(0\right)\sqrt{9}\right)\\
&=&\left(\left(-5\right)\sqrt{3}\right)+\left(\left(\dfrac{225}{7}\right)\sqrt{3}+\dfrac{195}{7}+\left(\dfrac{171}{5}\right)\sqrt{3}+\left(12\right)\sqrt{3}+\left(-26\right)\sqrt{3}+\left(-\dfrac{81}{2}\right)\sqrt{3}+4+0\right)\\
&=&\left(-5\right)\sqrt{3}+\left(\dfrac{225}{7}\right)\sqrt{3}+\dfrac{195}{7}+\left(\dfrac{171}{5}\right)\sqrt{3}+\left(12\right)\sqrt{3}+\left(-26\right)\sqrt{3}+\left(-\dfrac{81}{2}\right)\sqrt{3}+4+0\\
&=&\left(\dfrac{479}{70}\right)\sqrt{3}+\dfrac{223}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-1\right)\sqrt{75}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{27}+\left(\dfrac{65}{7}\right)\sqrt{9}+\left(\dfrac{57}{5}\right)\sqrt{27}+\left(4\right)\sqrt{27}+\left(-\dfrac{26}{3}\right)\sqrt{27}+\left(-\dfrac{81}{4}\right)\sqrt{12}+4+\left(0\right)\sqrt{9}\right)\\
&=&\left(\left(-5\right)\sqrt{3}\right)-\left(\left(\dfrac{225}{7}\right)\sqrt{3}+\dfrac{195}{7}+\left(\dfrac{171}{5}\right)\sqrt{3}+\left(12\right)\sqrt{3}+\left(-26\right)\sqrt{3}+\left(-\dfrac{81}{2}\right)\sqrt{3}+4+0\right)\\
&=&\left(\left(-5\right)\sqrt{3}\right)-\left(\left(\dfrac{829}{70}\right)\sqrt{3}+\dfrac{223}{7}\right)\\
&=&\left(-5\right)\sqrt{3}+\left(-\dfrac{829}{70}\right)\sqrt{3}-\dfrac{223}{7}\\
&=&\left(-\dfrac{1179}{70}\right)\sqrt{3}-\dfrac{223}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-1\right)\sqrt{75}\right)\times\left(\left(\dfrac{75}{7}\right)\sqrt{27}+\left(\dfrac{65}{7}\right)\sqrt{9}+\left(\dfrac{57}{5}\right)\sqrt{27}+\left(4\right)\sqrt{27}+\left(-\dfrac{26}{3}\right)\sqrt{27}+\left(-\dfrac{81}{4}\right)\sqrt{12}+4+\left(0\right)\sqrt{9}\right)\\
&=&\left(\left(-5\right)\sqrt{3}\right)\times\left(\left(\dfrac{225}{7}\right)\sqrt{3}+\dfrac{195}{7}+\left(\dfrac{171}{5}\right)\sqrt{3}+\left(12\right)\sqrt{3}+\left(-26\right)\sqrt{3}+\left(-\dfrac{81}{2}\right)\sqrt{3}+4+0\right)\\
&=&\left(\left(-5\right)\sqrt{3}\right)\left(\left(\dfrac{829}{70}\right)\sqrt{3}+\dfrac{223}{7}\right)\\
&=&\left(-\dfrac{829}{14}\right)\sqrt{9}+\left(-\dfrac{1115}{7}\right)\sqrt{3}\\
&=&-\dfrac{2487}{14}+\left(-\dfrac{1115}{7}\right)\sqrt{3}\\
\end{eqnarray*}