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 \( X=\left(\left(6\right)\sqrt{20}\right)-\dfrac{77}{5}+\dfrac{64}{5}+\left(\left(-7\right)\sqrt{20}\right)-\dfrac{15}{8}+\left(-\dfrac{61}{8}\right)\sqrt{20}+\left(-5\right)\sqrt{20}\) et \( Y=\left(\left(\left(\dfrac{39}{4}\right)\sqrt{20}\right)+\dfrac{3}{5}\right)-\dfrac{31}{9}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(6\right)\sqrt{20}\right)-\dfrac{77}{5}+\dfrac{64}{5}+\left(\left(-7\right)\sqrt{20}\right)-\dfrac{15}{8}+\left(-\dfrac{61}{8}\right)\sqrt{20}+\left(-5\right)\sqrt{20}\right)+\left(\left(\left(\left(\dfrac{39}{4}\right)\sqrt{20}\right)+\dfrac{3}{5}\right)-\dfrac{31}{9}\right)\\
&=&\left(\left(\left(12\right)\sqrt{5}\right)-\dfrac{77}{5}+\dfrac{64}{5}+\left(\left(-14\right)\sqrt{5}\right)-\dfrac{15}{8}+\left(-\dfrac{61}{4}\right)\sqrt{5}+\left(-10\right)\sqrt{5}\right)+\left(\left(\left(\left(\dfrac{39}{2}\right)\sqrt{5}\right)+\dfrac{3}{5}\right)-\dfrac{31}{9}\right)\\
&=&\left(\left(12\right)\sqrt{5}\right)-\dfrac{77}{5}+\dfrac{64}{5}+\left(\left(-14\right)\sqrt{5}\right)-\dfrac{15}{8}+\left(-\dfrac{61}{4}\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\left(\left(\left(\dfrac{39}{2}\right)\sqrt{5}\right)+\dfrac{3}{5}\right)-\dfrac{31}{9}\\
&=&\left(-\dfrac{31}{4}\right)\sqrt{5}-\dfrac{527}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(6\right)\sqrt{20}\right)-\dfrac{77}{5}+\dfrac{64}{5}+\left(\left(-7\right)\sqrt{20}\right)-\dfrac{15}{8}+\left(-\dfrac{61}{8}\right)\sqrt{20}+\left(-5\right)\sqrt{20}\right)-\left(\left(\left(\left(\dfrac{39}{4}\right)\sqrt{20}\right)+\dfrac{3}{5}\right)-\dfrac{31}{9}\right)\\
&=&\left(\left(\left(12\right)\sqrt{5}\right)-\dfrac{77}{5}+\dfrac{64}{5}+\left(\left(-14\right)\sqrt{5}\right)-\dfrac{15}{8}+\left(-\dfrac{61}{4}\right)\sqrt{5}+\left(-10\right)\sqrt{5}\right)-\left(\left(\left(\left(\dfrac{39}{2}\right)\sqrt{5}\right)+\dfrac{3}{5}\right)-\dfrac{31}{9}\right)\\
&=&\left(\left(-\dfrac{109}{4}\right)\sqrt{5}-\dfrac{179}{40}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{5}-\dfrac{128}{45}\right)\\
&=&\left(-\dfrac{109}{4}\right)\sqrt{5}-\dfrac{179}{40}+\left(-\dfrac{39}{2}\right)\sqrt{5}+\dfrac{128}{45}\\
&=&\left(-\dfrac{187}{4}\right)\sqrt{5}-\dfrac{587}{360}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(6\right)\sqrt{20}\right)-\dfrac{77}{5}+\dfrac{64}{5}+\left(\left(-7\right)\sqrt{20}\right)-\dfrac{15}{8}+\left(-\dfrac{61}{8}\right)\sqrt{20}+\left(-5\right)\sqrt{20}\right)\times\left(\left(\left(\left(\dfrac{39}{4}\right)\sqrt{20}\right)+\dfrac{3}{5}\right)-\dfrac{31}{9}\right)\\
&=&\left(\left(\left(12\right)\sqrt{5}\right)-\dfrac{77}{5}+\dfrac{64}{5}+\left(\left(-14\right)\sqrt{5}\right)-\dfrac{15}{8}+\left(-\dfrac{61}{4}\right)\sqrt{5}+\left(-10\right)\sqrt{5}\right)\times\left(\left(\left(\left(\dfrac{39}{2}\right)\sqrt{5}\right)+\dfrac{3}{5}\right)-\dfrac{31}{9}\right)\\
&=&\left(\left(-\dfrac{109}{4}\right)\sqrt{5}-\dfrac{179}{40}\right)\left(\left(\dfrac{39}{2}\right)\sqrt{5}-\dfrac{128}{45}\right)\\
&=&\left(-\dfrac{4251}{8}\right)\sqrt{25}+\left(-\dfrac{7021}{720}\right)\sqrt{5}+\dfrac{2864}{225}\\
&=&-\dfrac{4759463}{1800}+\left(-\dfrac{7021}{720}\right)\sqrt{5}\\
\end{eqnarray*}