L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{7}{2}+\dfrac{63}{5}+\left(-\dfrac{6}{7}\right)\sqrt{20}-\dfrac{62}{9}+\left(\left(-\dfrac{27}{7}\right)\sqrt{25}\right)-\left(\left(\dfrac{29}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{125}\right)+\left(-\dfrac{22}{9}\right)\sqrt{20}+\left(-\dfrac{77}{5}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{25}\) et \( Y=\left(\left(\dfrac{3}{2}\right)\sqrt{125}\right)-\left(\left(\left(-5\right)\sqrt{20}\right)+9-\left(\left(\dfrac{59}{7}\right)\sqrt{125}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{7}{2}+\dfrac{63}{5}+\left(-\dfrac{6}{7}\right)\sqrt{20}-\dfrac{62}{9}+\left(\left(-\dfrac{27}{7}\right)\sqrt{25}\right)-\left(\left(\dfrac{29}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{125}\right)+\left(-\dfrac{22}{9}\right)\sqrt{20}+\left(-\dfrac{77}{5}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{25}\right)+\left(\left(\left(\dfrac{3}{2}\right)\sqrt{125}\right)-\left(\left(\left(-5\right)\sqrt{20}\right)+9-\left(\left(\dfrac{59}{7}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(-\dfrac{7}{2}+\dfrac{63}{5}+\left(-\dfrac{12}{7}\right)\sqrt{5}-\dfrac{62}{9}-\dfrac{135}{7}-\dfrac{145}{3}-\left(\left(-\dfrac{200}{3}\right)\sqrt{5}\right)+\left(-\dfrac{44}{9}\right)\sqrt{5}+\left(-77\right)\sqrt{5}+\dfrac{75}{2}\right)+\left(\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(\left(-10\right)\sqrt{5}\right)+9-\left(\left(\dfrac{295}{7}\right)\sqrt{5}\right)\right)\right)\\
&=&-\dfrac{7}{2}+\dfrac{63}{5}+\left(-\dfrac{12}{7}\right)\sqrt{5}-\dfrac{62}{9}-\dfrac{135}{7}-\dfrac{145}{3}-\left(\left(-\dfrac{200}{3}\right)\sqrt{5}\right)+\left(-\dfrac{44}{9}\right)\sqrt{5}+\left(-77\right)\sqrt{5}+\dfrac{75}{2}+\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(\left(-10\right)\sqrt{5}\right)+9-\left(\left(\dfrac{295}{7}\right)\sqrt{5}\right)\right)\\
&=&-\dfrac{11626}{315}+\left(\dfrac{5381}{126}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{7}{2}+\dfrac{63}{5}+\left(-\dfrac{6}{7}\right)\sqrt{20}-\dfrac{62}{9}+\left(\left(-\dfrac{27}{7}\right)\sqrt{25}\right)-\left(\left(\dfrac{29}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{125}\right)+\left(-\dfrac{22}{9}\right)\sqrt{20}+\left(-\dfrac{77}{5}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{3}{2}\right)\sqrt{125}\right)-\left(\left(\left(-5\right)\sqrt{20}\right)+9-\left(\left(\dfrac{59}{7}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(-\dfrac{7}{2}+\dfrac{63}{5}+\left(-\dfrac{12}{7}\right)\sqrt{5}-\dfrac{62}{9}-\dfrac{135}{7}-\dfrac{145}{3}-\left(\left(-\dfrac{200}{3}\right)\sqrt{5}\right)+\left(-\dfrac{44}{9}\right)\sqrt{5}+\left(-77\right)\sqrt{5}+\dfrac{75}{2}\right)-\left(\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(\left(-10\right)\sqrt{5}\right)+9-\left(\left(\dfrac{295}{7}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(-\dfrac{8791}{315}+\left(-\dfrac{1067}{63}\right)\sqrt{5}\right)-\left(\left(\dfrac{835}{14}\right)\sqrt{5}-9\right)\\
&=&-\dfrac{8791}{315}+\left(-\dfrac{1067}{63}\right)\sqrt{5}+\left(-\dfrac{835}{14}\right)\sqrt{5}+9\\
&=&-\dfrac{5956}{315}+\left(-\dfrac{9649}{126}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{7}{2}+\dfrac{63}{5}+\left(-\dfrac{6}{7}\right)\sqrt{20}-\dfrac{62}{9}+\left(\left(-\dfrac{27}{7}\right)\sqrt{25}\right)-\left(\left(\dfrac{29}{3}\right)\sqrt{25}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{125}\right)+\left(-\dfrac{22}{9}\right)\sqrt{20}+\left(-\dfrac{77}{5}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{25}\right)\times\left(\left(\left(\dfrac{3}{2}\right)\sqrt{125}\right)-\left(\left(\left(-5\right)\sqrt{20}\right)+9-\left(\left(\dfrac{59}{7}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(-\dfrac{7}{2}+\dfrac{63}{5}+\left(-\dfrac{12}{7}\right)\sqrt{5}-\dfrac{62}{9}-\dfrac{135}{7}-\dfrac{145}{3}-\left(\left(-\dfrac{200}{3}\right)\sqrt{5}\right)+\left(-\dfrac{44}{9}\right)\sqrt{5}+\left(-77\right)\sqrt{5}+\dfrac{75}{2}\right)\times\left(\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(\left(-10\right)\sqrt{5}\right)+9-\left(\left(\dfrac{295}{7}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(-\dfrac{8791}{315}+\left(-\dfrac{1067}{63}\right)\sqrt{5}\right)\left(\left(\dfrac{835}{14}\right)\sqrt{5}-9\right)\\
&=&\left(-\dfrac{1333655}{882}\right)\sqrt{5}+\dfrac{8791}{35}+\left(-\dfrac{890945}{882}\right)\sqrt{25}\\
&=&\left(-\dfrac{1333655}{882}\right)\sqrt{5}-\dfrac{21165959}{4410}\\
\end{eqnarray*}