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 \( X=\left(\left(-\dfrac{76}{7}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{2}{3}\right)\sqrt{20}\right)-\left(\left(0\right)\sqrt{20}\right)-\left(\left(-\dfrac{44}{7}\right)\sqrt{45}\right)\right)\) et \( Y=\left(\dfrac{47}{5}\right)\sqrt{45}-\dfrac{42}{5}+\dfrac{50}{3}-\left(\left(-\dfrac{75}{8}\right)\sqrt{125}\right)+\left(\left(\dfrac{43}{2}\right)\sqrt{25}\right)-\left(\left(-8\right)\sqrt{125}\right)\) . 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{76}{7}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{2}{3}\right)\sqrt{20}\right)-\left(\left(0\right)\sqrt{20}\right)-\left(\left(-\dfrac{44}{7}\right)\sqrt{45}\right)\right)\right)+\left(\left(\dfrac{47}{5}\right)\sqrt{45}-\dfrac{42}{5}+\dfrac{50}{3}-\left(\left(-\dfrac{75}{8}\right)\sqrt{125}\right)+\left(\left(\dfrac{43}{2}\right)\sqrt{25}\right)-\left(\left(-8\right)\sqrt{125}\right)\right)\\
&=&\left(-\dfrac{380}{7}-\left(\left(\left(\dfrac{4}{3}\right)\sqrt{5}\right)-\left(\left(0\right)\sqrt{5}\right)-\left(\left(-\dfrac{132}{7}\right)\sqrt{5}\right)\right)\right)+\left(\left(\dfrac{141}{5}\right)\sqrt{5}-\dfrac{42}{5}+\dfrac{50}{3}-\left(\left(-\dfrac{375}{8}\right)\sqrt{5}\right)+\dfrac{215}{2}-\left(\left(-40\right)\sqrt{5}\right)\right)\\
&=&-\dfrac{380}{7}-\left(\left(\left(\dfrac{4}{3}\right)\sqrt{5}\right)-\left(\left(0\right)\sqrt{5}\right)-\left(\left(-\dfrac{132}{7}\right)\sqrt{5}\right)\right)+\left(\dfrac{141}{5}\right)\sqrt{5}-\dfrac{42}{5}+\dfrac{50}{3}-\left(\left(-\dfrac{375}{8}\right)\sqrt{5}\right)+\dfrac{215}{2}-\left(\left(-40\right)\sqrt{5}\right)\\
&=&\dfrac{12911}{210}+\left(\dfrac{79703}{840}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{76}{7}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{2}{3}\right)\sqrt{20}\right)-\left(\left(0\right)\sqrt{20}\right)-\left(\left(-\dfrac{44}{7}\right)\sqrt{45}\right)\right)\right)-\left(\left(\dfrac{47}{5}\right)\sqrt{45}-\dfrac{42}{5}+\dfrac{50}{3}-\left(\left(-\dfrac{75}{8}\right)\sqrt{125}\right)+\left(\left(\dfrac{43}{2}\right)\sqrt{25}\right)-\left(\left(-8\right)\sqrt{125}\right)\right)\\
&=&\left(-\dfrac{380}{7}-\left(\left(\left(\dfrac{4}{3}\right)\sqrt{5}\right)-\left(\left(0\right)\sqrt{5}\right)-\left(\left(-\dfrac{132}{7}\right)\sqrt{5}\right)\right)\right)-\left(\left(\dfrac{141}{5}\right)\sqrt{5}-\dfrac{42}{5}+\dfrac{50}{3}-\left(\left(-\dfrac{375}{8}\right)\sqrt{5}\right)+\dfrac{215}{2}-\left(\left(-40\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{380}{7}+\left(-\dfrac{424}{21}\right)\sqrt{5}\right)-\left(\left(\dfrac{4603}{40}\right)\sqrt{5}+\dfrac{3473}{30}\right)\\
&=&-\dfrac{380}{7}+\left(-\dfrac{424}{21}\right)\sqrt{5}+\left(-\dfrac{4603}{40}\right)\sqrt{5}-\dfrac{3473}{30}\\
&=&-\dfrac{35711}{210}+\left(-\dfrac{113623}{840}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{76}{7}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{2}{3}\right)\sqrt{20}\right)-\left(\left(0\right)\sqrt{20}\right)-\left(\left(-\dfrac{44}{7}\right)\sqrt{45}\right)\right)\right)\times\left(\left(\dfrac{47}{5}\right)\sqrt{45}-\dfrac{42}{5}+\dfrac{50}{3}-\left(\left(-\dfrac{75}{8}\right)\sqrt{125}\right)+\left(\left(\dfrac{43}{2}\right)\sqrt{25}\right)-\left(\left(-8\right)\sqrt{125}\right)\right)\\
&=&\left(-\dfrac{380}{7}-\left(\left(\left(\dfrac{4}{3}\right)\sqrt{5}\right)-\left(\left(0\right)\sqrt{5}\right)-\left(\left(-\dfrac{132}{7}\right)\sqrt{5}\right)\right)\right)\times\left(\left(\dfrac{141}{5}\right)\sqrt{5}-\dfrac{42}{5}+\dfrac{50}{3}-\left(\left(-\dfrac{375}{8}\right)\sqrt{5}\right)+\dfrac{215}{2}-\left(\left(-40\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{380}{7}+\left(-\dfrac{424}{21}\right)\sqrt{5}\right)\left(\left(\dfrac{4603}{40}\right)\sqrt{5}+\dfrac{3473}{30}\right)\\
&=&\left(-\dfrac{5408117}{630}\right)\sqrt{5}-\dfrac{131974}{21}+\left(-\dfrac{243959}{105}\right)\sqrt{25}\\
&=&\left(-\dfrac{5408117}{630}\right)\sqrt{5}-\dfrac{125311}{7}\\
\end{eqnarray*}