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 \( X=-\dfrac{46}{5}-\left(\left(-7\right)\sqrt{25}\right)+\left(-\dfrac{23}{2}\right)\sqrt{25}+\left(\left(\dfrac{3}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{20}\right)+\left(-1\right)\sqrt{20}+\left(7\right)\sqrt{20}+\left(-1\right)\sqrt{20}+\dfrac{29}{4}\) et \( Y=\left(\left(-\dfrac{57}{2}\right)\sqrt{45}+\left(-\dfrac{7}{3}\right)\sqrt{125}+\left(\dfrac{15}{4}\right)\sqrt{20}\right)-\left(\left(\left(3\right)\sqrt{25}\right)-\left(\left(-\dfrac{22}{7}\right)\sqrt{125}\right)+\dfrac{69}{8}\right)-\left(\left(4\right)\sqrt{20}+\left(-\dfrac{37}{9}\right)\sqrt{25}+\left(3\right)\sqrt{25}+\left(0\right)\sqrt{20}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{46}{5}-\left(\left(-7\right)\sqrt{25}\right)+\left(-\dfrac{23}{2}\right)\sqrt{25}+\left(\left(\dfrac{3}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{20}\right)+\left(-1\right)\sqrt{20}+\left(7\right)\sqrt{20}+\left(-1\right)\sqrt{20}+\dfrac{29}{4}\right)+\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{45}+\left(-\dfrac{7}{3}\right)\sqrt{125}+\left(\dfrac{15}{4}\right)\sqrt{20}\right)-\left(\left(\left(3\right)\sqrt{25}\right)-\left(\left(-\dfrac{22}{7}\right)\sqrt{125}\right)+\dfrac{69}{8}\right)-\left(\left(4\right)\sqrt{20}+\left(-\dfrac{37}{9}\right)\sqrt{25}+\left(3\right)\sqrt{25}+\left(0\right)\sqrt{20}\right)\right)\\
&=&\left(-\dfrac{46}{5}+35-\dfrac{115}{2}+\left(\left(3\right)\sqrt{5}\right)-\left(\left(3\right)\sqrt{5}\right)+\left(-2\right)\sqrt{5}+\left(14\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\dfrac{29}{4}\right)+\left(\left(\left(-\dfrac{171}{2}\right)\sqrt{5}+\left(-\dfrac{35}{3}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(15-\left(\left(-\dfrac{110}{7}\right)\sqrt{5}\right)+\dfrac{69}{8}\right)-\left(\left(8\right)\sqrt{5}-\dfrac{185}{9}+15+\left(0\right)\sqrt{5}\right)\right)\\
&=&-\dfrac{46}{5}+35-\dfrac{115}{2}+\left(\left(3\right)\sqrt{5}\right)-\left(\left(3\right)\sqrt{5}\right)+\left(-2\right)\sqrt{5}+\left(14\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\dfrac{29}{4}+\left(\left(-\dfrac{171}{2}\right)\sqrt{5}+\left(-\dfrac{35}{3}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(15-\left(\left(-\dfrac{110}{7}\right)\sqrt{5}\right)+\dfrac{69}{8}\right)-\left(\left(8\right)\sqrt{5}-\dfrac{185}{9}+15+\left(0\right)\sqrt{5}\right)\\
&=&-\dfrac{15307}{360}+\left(-\dfrac{2171}{21}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{46}{5}-\left(\left(-7\right)\sqrt{25}\right)+\left(-\dfrac{23}{2}\right)\sqrt{25}+\left(\left(\dfrac{3}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{20}\right)+\left(-1\right)\sqrt{20}+\left(7\right)\sqrt{20}+\left(-1\right)\sqrt{20}+\dfrac{29}{4}\right)-\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{45}+\left(-\dfrac{7}{3}\right)\sqrt{125}+\left(\dfrac{15}{4}\right)\sqrt{20}\right)-\left(\left(\left(3\right)\sqrt{25}\right)-\left(\left(-\dfrac{22}{7}\right)\sqrt{125}\right)+\dfrac{69}{8}\right)-\left(\left(4\right)\sqrt{20}+\left(-\dfrac{37}{9}\right)\sqrt{25}+\left(3\right)\sqrt{25}+\left(0\right)\sqrt{20}\right)\right)\\
&=&\left(-\dfrac{46}{5}+35-\dfrac{115}{2}+\left(\left(3\right)\sqrt{5}\right)-\left(\left(3\right)\sqrt{5}\right)+\left(-2\right)\sqrt{5}+\left(14\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\dfrac{29}{4}\right)-\left(\left(\left(-\dfrac{171}{2}\right)\sqrt{5}+\left(-\dfrac{35}{3}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(15-\left(\left(-\dfrac{110}{7}\right)\sqrt{5}\right)+\dfrac{69}{8}\right)-\left(\left(8\right)\sqrt{5}-\dfrac{185}{9}+15+\left(0\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{489}{20}+\left(10\right)\sqrt{5}\right)-\left(\left(-\dfrac{2381}{21}\right)\sqrt{5}-\dfrac{1301}{72}\right)\\
&=&-\dfrac{489}{20}+\left(10\right)\sqrt{5}+\left(\dfrac{2381}{21}\right)\sqrt{5}+\dfrac{1301}{72}\\
&=&-\dfrac{2297}{360}+\left(\dfrac{2591}{21}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{46}{5}-\left(\left(-7\right)\sqrt{25}\right)+\left(-\dfrac{23}{2}\right)\sqrt{25}+\left(\left(\dfrac{3}{2}\right)\sqrt{20}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{20}\right)+\left(-1\right)\sqrt{20}+\left(7\right)\sqrt{20}+\left(-1\right)\sqrt{20}+\dfrac{29}{4}\right)\times\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{45}+\left(-\dfrac{7}{3}\right)\sqrt{125}+\left(\dfrac{15}{4}\right)\sqrt{20}\right)-\left(\left(\left(3\right)\sqrt{25}\right)-\left(\left(-\dfrac{22}{7}\right)\sqrt{125}\right)+\dfrac{69}{8}\right)-\left(\left(4\right)\sqrt{20}+\left(-\dfrac{37}{9}\right)\sqrt{25}+\left(3\right)\sqrt{25}+\left(0\right)\sqrt{20}\right)\right)\\
&=&\left(-\dfrac{46}{5}+35-\dfrac{115}{2}+\left(\left(3\right)\sqrt{5}\right)-\left(\left(3\right)\sqrt{5}\right)+\left(-2\right)\sqrt{5}+\left(14\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\dfrac{29}{4}\right)\times\left(\left(\left(-\dfrac{171}{2}\right)\sqrt{5}+\left(-\dfrac{35}{3}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(15-\left(\left(-\dfrac{110}{7}\right)\sqrt{5}\right)+\dfrac{69}{8}\right)-\left(\left(8\right)\sqrt{5}-\dfrac{185}{9}+15+\left(0\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{489}{20}+\left(10\right)\sqrt{5}\right)\left(\left(-\dfrac{2381}{21}\right)\sqrt{5}-\dfrac{1301}{72}\right)\\
&=&\left(\dfrac{816313}{315}\right)\sqrt{5}+\dfrac{212063}{480}+\left(-\dfrac{23810}{21}\right)\sqrt{25}\\
&=&\left(\dfrac{816313}{315}\right)\sqrt{5}-\dfrac{17563559}{3360}\\
\end{eqnarray*}