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 \( X=\left(\left(6\right)\sqrt{8}+\left(-4\right)\sqrt{18}-4+\left(-\dfrac{59}{9}\right)\sqrt{50}\right)-\left(\left(-4\right)\sqrt{18}\right)-\left(\dfrac{29}{2}-\left(\left(\dfrac{32}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{18}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{50}\right)\right)\) et \( Y=\left(\dfrac{79}{4}\right)\sqrt{8}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(6\right)\sqrt{8}+\left(-4\right)\sqrt{18}-4+\left(-\dfrac{59}{9}\right)\sqrt{50}\right)-\left(\left(-4\right)\sqrt{18}\right)-\left(\dfrac{29}{2}-\left(\left(\dfrac{32}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{18}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{50}\right)\right)\right)+\left(\left(\dfrac{79}{4}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(12\right)\sqrt{2}+\left(-12\right)\sqrt{2}-4+\left(-\dfrac{295}{9}\right)\sqrt{2}\right)-\left(\left(-12\right)\sqrt{2}\right)-\left(\dfrac{29}{2}-\left(\left(\dfrac{96}{5}\right)\sqrt{2}\right)+33-\left(\left(24\right)\sqrt{2}\right)-\left(\left(-\dfrac{85}{4}\right)\sqrt{2}\right)\right)\right)+\left(\left(\dfrac{79}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(12\right)\sqrt{2}+\left(-12\right)\sqrt{2}-4+\left(-\dfrac{295}{9}\right)\sqrt{2}\right)-\left(\left(-12\right)\sqrt{2}\right)-\left(\dfrac{29}{2}-\left(\left(\dfrac{96}{5}\right)\sqrt{2}\right)+33-\left(\left(24\right)\sqrt{2}\right)-\left(\left(-\dfrac{85}{4}\right)\sqrt{2}\right)\right)+\left(\dfrac{79}{2}\right)\sqrt{2}\\
&=&\left(\dfrac{7321}{180}\right)\sqrt{2}-\dfrac{103}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(6\right)\sqrt{8}+\left(-4\right)\sqrt{18}-4+\left(-\dfrac{59}{9}\right)\sqrt{50}\right)-\left(\left(-4\right)\sqrt{18}\right)-\left(\dfrac{29}{2}-\left(\left(\dfrac{32}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{18}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{50}\right)\right)\right)-\left(\left(\dfrac{79}{4}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(12\right)\sqrt{2}+\left(-12\right)\sqrt{2}-4+\left(-\dfrac{295}{9}\right)\sqrt{2}\right)-\left(\left(-12\right)\sqrt{2}\right)-\left(\dfrac{29}{2}-\left(\left(\dfrac{96}{5}\right)\sqrt{2}\right)+33-\left(\left(24\right)\sqrt{2}\right)-\left(\left(-\dfrac{85}{4}\right)\sqrt{2}\right)\right)\right)-\left(\left(\dfrac{79}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{211}{180}\right)\sqrt{2}-\dfrac{103}{2}\right)-\left(\left(\dfrac{79}{2}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{211}{180}\right)\sqrt{2}-\dfrac{103}{2}+\left(-\dfrac{79}{2}\right)\sqrt{2}\\
&=&\left(-\dfrac{6899}{180}\right)\sqrt{2}-\dfrac{103}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(6\right)\sqrt{8}+\left(-4\right)\sqrt{18}-4+\left(-\dfrac{59}{9}\right)\sqrt{50}\right)-\left(\left(-4\right)\sqrt{18}\right)-\left(\dfrac{29}{2}-\left(\left(\dfrac{32}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{18}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{50}\right)\right)\right)\times\left(\left(\dfrac{79}{4}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(12\right)\sqrt{2}+\left(-12\right)\sqrt{2}-4+\left(-\dfrac{295}{9}\right)\sqrt{2}\right)-\left(\left(-12\right)\sqrt{2}\right)-\left(\dfrac{29}{2}-\left(\left(\dfrac{96}{5}\right)\sqrt{2}\right)+33-\left(\left(24\right)\sqrt{2}\right)-\left(\left(-\dfrac{85}{4}\right)\sqrt{2}\right)\right)\right)\times\left(\left(\dfrac{79}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{211}{180}\right)\sqrt{2}-\dfrac{103}{2}\right)\left(\left(\dfrac{79}{2}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{16669}{360}\right)\sqrt{4}+\left(-\dfrac{8137}{4}\right)\sqrt{2}\\
&=&\dfrac{16669}{180}+\left(-\dfrac{8137}{4}\right)\sqrt{2}\\
\end{eqnarray*}