L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{58}{5}-\left(\left(1\right)\sqrt{49}\right)\right)-\left(-\dfrac{1}{9}+\dfrac{58}{5}\right)-\left(\left(\left(8\right)\sqrt{63}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{63}\right)\right)\) et \( Y=-\dfrac{40}{9}+\left(-\dfrac{4}{3}\right)\sqrt{175}+\left(3\right)\sqrt{175}+\left(\dfrac{15}{2}\right)\sqrt{28}+\left(0\right)\sqrt{63}+\left(\dfrac{38}{5}\right)\sqrt{175}+\left(\dfrac{17}{9}\right)\sqrt{175}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{58}{5}-\left(\left(1\right)\sqrt{49}\right)\right)-\left(-\dfrac{1}{9}+\dfrac{58}{5}\right)-\left(\left(\left(8\right)\sqrt{63}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{63}\right)\right)\right)+\left(-\dfrac{40}{9}+\left(-\dfrac{4}{3}\right)\sqrt{175}+\left(3\right)\sqrt{175}+\left(\dfrac{15}{2}\right)\sqrt{28}+\left(0\right)\sqrt{63}+\left(\dfrac{38}{5}\right)\sqrt{175}+\left(\dfrac{17}{9}\right)\sqrt{175}\right)\\
&=&\left(\left(-\dfrac{58}{5}-7\right)-\left(-\dfrac{1}{9}+\dfrac{58}{5}\right)-\left(\left(\left(24\right)\sqrt{7}\right)-\left(\left(\dfrac{93}{4}\right)\sqrt{7}\right)\right)\right)+\left(-\dfrac{40}{9}+\left(-\dfrac{20}{3}\right)\sqrt{7}+\left(15\right)\sqrt{7}+\left(15\right)\sqrt{7}+\left(0\right)\sqrt{7}+\left(38\right)\sqrt{7}+\left(\dfrac{85}{9}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{58}{5}-7\right)-\left(-\dfrac{1}{9}+\dfrac{58}{5}\right)-\left(\left(\left(24\right)\sqrt{7}\right)-\left(\left(\dfrac{93}{4}\right)\sqrt{7}\right)\right)-\dfrac{40}{9}+\left(-\dfrac{20}{3}\right)\sqrt{7}+\left(15\right)\sqrt{7}+\left(15\right)\sqrt{7}+\left(0\right)\sqrt{7}+\left(38\right)\sqrt{7}+\left(\dfrac{85}{9}\right)\sqrt{7}\\
&=&-\dfrac{518}{15}+\left(\dfrac{2521}{36}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{58}{5}-\left(\left(1\right)\sqrt{49}\right)\right)-\left(-\dfrac{1}{9}+\dfrac{58}{5}\right)-\left(\left(\left(8\right)\sqrt{63}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{63}\right)\right)\right)-\left(-\dfrac{40}{9}+\left(-\dfrac{4}{3}\right)\sqrt{175}+\left(3\right)\sqrt{175}+\left(\dfrac{15}{2}\right)\sqrt{28}+\left(0\right)\sqrt{63}+\left(\dfrac{38}{5}\right)\sqrt{175}+\left(\dfrac{17}{9}\right)\sqrt{175}\right)\\
&=&\left(\left(-\dfrac{58}{5}-7\right)-\left(-\dfrac{1}{9}+\dfrac{58}{5}\right)-\left(\left(\left(24\right)\sqrt{7}\right)-\left(\left(\dfrac{93}{4}\right)\sqrt{7}\right)\right)\right)-\left(-\dfrac{40}{9}+\left(-\dfrac{20}{3}\right)\sqrt{7}+\left(15\right)\sqrt{7}+\left(15\right)\sqrt{7}+\left(0\right)\sqrt{7}+\left(38\right)\sqrt{7}+\left(\dfrac{85}{9}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{1354}{45}+\left(-\dfrac{3}{4}\right)\sqrt{7}\right)-\left(-\dfrac{40}{9}+\left(\dfrac{637}{9}\right)\sqrt{7}\right)\\
&=&-\dfrac{1354}{45}+\left(-\dfrac{3}{4}\right)\sqrt{7}+\dfrac{40}{9}+\left(-\dfrac{637}{9}\right)\sqrt{7}\\
&=&-\dfrac{1154}{45}+\left(-\dfrac{2575}{36}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{58}{5}-\left(\left(1\right)\sqrt{49}\right)\right)-\left(-\dfrac{1}{9}+\dfrac{58}{5}\right)-\left(\left(\left(8\right)\sqrt{63}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{63}\right)\right)\right)\times\left(-\dfrac{40}{9}+\left(-\dfrac{4}{3}\right)\sqrt{175}+\left(3\right)\sqrt{175}+\left(\dfrac{15}{2}\right)\sqrt{28}+\left(0\right)\sqrt{63}+\left(\dfrac{38}{5}\right)\sqrt{175}+\left(\dfrac{17}{9}\right)\sqrt{175}\right)\\
&=&\left(\left(-\dfrac{58}{5}-7\right)-\left(-\dfrac{1}{9}+\dfrac{58}{5}\right)-\left(\left(\left(24\right)\sqrt{7}\right)-\left(\left(\dfrac{93}{4}\right)\sqrt{7}\right)\right)\right)\times\left(-\dfrac{40}{9}+\left(-\dfrac{20}{3}\right)\sqrt{7}+\left(15\right)\sqrt{7}+\left(15\right)\sqrt{7}+\left(0\right)\sqrt{7}+\left(38\right)\sqrt{7}+\left(\dfrac{85}{9}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{1354}{45}+\left(-\dfrac{3}{4}\right)\sqrt{7}\right)\left(-\dfrac{40}{9}+\left(\dfrac{637}{9}\right)\sqrt{7}\right)\\
&=&\dfrac{10832}{81}+\left(-\dfrac{861148}{405}\right)\sqrt{7}+\left(-\dfrac{637}{12}\right)\sqrt{49}\\
&=&-\dfrac{77065}{324}+\left(-\dfrac{861148}{405}\right)\sqrt{7}\\
\end{eqnarray*}