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 \( X=\left(\left(-\dfrac{4}{3}\right)\sqrt{4}+\left(-\dfrac{21}{8}\right)\sqrt{18}+\left(\dfrac{31}{4}\right)\sqrt{8}\right)-\left(\left(\left(6\right)\sqrt{4}\right)+\dfrac{80}{9}-\left(\left(-9\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{5}\right)\sqrt{18}\right)\right)-\left(-2+\dfrac{65}{9}+\left(\dfrac{45}{4}\right)\sqrt{18}+\left(-\dfrac{1}{5}\right)\sqrt{18}\right)\) et \( Y=\left(\left(-\dfrac{11}{6}\right)\sqrt{18}\right)-\left(\left(-\dfrac{74}{9}\right)\sqrt{4}\right)\) . 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(-\dfrac{4}{3}\right)\sqrt{4}+\left(-\dfrac{21}{8}\right)\sqrt{18}+\left(\dfrac{31}{4}\right)\sqrt{8}\right)-\left(\left(\left(6\right)\sqrt{4}\right)+\dfrac{80}{9}-\left(\left(-9\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{5}\right)\sqrt{18}\right)\right)-\left(-2+\dfrac{65}{9}+\left(\dfrac{45}{4}\right)\sqrt{18}+\left(-\dfrac{1}{5}\right)\sqrt{18}\right)\right)+\left(\left(\left(-\dfrac{11}{6}\right)\sqrt{18}\right)-\left(\left(-\dfrac{74}{9}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(-\dfrac{8}{3}+\left(-\dfrac{63}{8}\right)\sqrt{2}+\left(\dfrac{31}{2}\right)\sqrt{2}\right)-\left(12+\dfrac{80}{9}-\left(\left(-18\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{5}\right)\sqrt{2}\right)\right)-\left(-2+\dfrac{65}{9}+\left(\dfrac{135}{4}\right)\sqrt{2}+\left(-\dfrac{3}{5}\right)\sqrt{2}\right)\right)+\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)+\dfrac{148}{9}\right)\\
&=&\left(-\dfrac{8}{3}+\left(-\dfrac{63}{8}\right)\sqrt{2}+\left(\dfrac{31}{2}\right)\sqrt{2}\right)-\left(12+\dfrac{80}{9}-\left(\left(-18\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{5}\right)\sqrt{2}\right)\right)-\left(-2+\dfrac{65}{9}+\left(\dfrac{135}{4}\right)\sqrt{2}+\left(-\dfrac{3}{5}\right)\sqrt{2}\right)+\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)+\dfrac{148}{9}\\
&=&-\dfrac{37}{3}+\left(-\dfrac{397}{8}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{4}{3}\right)\sqrt{4}+\left(-\dfrac{21}{8}\right)\sqrt{18}+\left(\dfrac{31}{4}\right)\sqrt{8}\right)-\left(\left(\left(6\right)\sqrt{4}\right)+\dfrac{80}{9}-\left(\left(-9\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{5}\right)\sqrt{18}\right)\right)-\left(-2+\dfrac{65}{9}+\left(\dfrac{45}{4}\right)\sqrt{18}+\left(-\dfrac{1}{5}\right)\sqrt{18}\right)\right)-\left(\left(\left(-\dfrac{11}{6}\right)\sqrt{18}\right)-\left(\left(-\dfrac{74}{9}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(-\dfrac{8}{3}+\left(-\dfrac{63}{8}\right)\sqrt{2}+\left(\dfrac{31}{2}\right)\sqrt{2}\right)-\left(12+\dfrac{80}{9}-\left(\left(-18\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{5}\right)\sqrt{2}\right)\right)-\left(-2+\dfrac{65}{9}+\left(\dfrac{135}{4}\right)\sqrt{2}+\left(-\dfrac{3}{5}\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)+\dfrac{148}{9}\right)\\
&=&\left(-\dfrac{259}{9}+\left(-\dfrac{353}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{2}+\dfrac{148}{9}\right)\\
&=&-\dfrac{259}{9}+\left(-\dfrac{353}{8}\right)\sqrt{2}+\left(\dfrac{11}{2}\right)\sqrt{2}-\dfrac{148}{9}\\
&=&-\dfrac{407}{9}+\left(-\dfrac{309}{8}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{4}{3}\right)\sqrt{4}+\left(-\dfrac{21}{8}\right)\sqrt{18}+\left(\dfrac{31}{4}\right)\sqrt{8}\right)-\left(\left(\left(6\right)\sqrt{4}\right)+\dfrac{80}{9}-\left(\left(-9\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{5}\right)\sqrt{18}\right)\right)-\left(-2+\dfrac{65}{9}+\left(\dfrac{45}{4}\right)\sqrt{18}+\left(-\dfrac{1}{5}\right)\sqrt{18}\right)\right)\times\left(\left(\left(-\dfrac{11}{6}\right)\sqrt{18}\right)-\left(\left(-\dfrac{74}{9}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(-\dfrac{8}{3}+\left(-\dfrac{63}{8}\right)\sqrt{2}+\left(\dfrac{31}{2}\right)\sqrt{2}\right)-\left(12+\dfrac{80}{9}-\left(\left(-18\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{5}\right)\sqrt{2}\right)\right)-\left(-2+\dfrac{65}{9}+\left(\dfrac{135}{4}\right)\sqrt{2}+\left(-\dfrac{3}{5}\right)\sqrt{2}\right)\right)\times\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)+\dfrac{148}{9}\right)\\
&=&\left(-\dfrac{259}{9}+\left(-\dfrac{353}{8}\right)\sqrt{2}\right)\left(\left(-\dfrac{11}{2}\right)\sqrt{2}+\dfrac{148}{9}\right)\\
&=&\left(-\dfrac{1702}{3}\right)\sqrt{2}-\dfrac{38332}{81}+\left(\dfrac{3883}{16}\right)\sqrt{4}\\
&=&\left(-\dfrac{1702}{3}\right)\sqrt{2}+\dfrac{7867}{648}\\
\end{eqnarray*}