L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{25}{8}\right)\sqrt{49}\) et \( Y=\left(\left(\dfrac{16}{3}\right)\sqrt{28}\right)-\left(\left(-7\right)\sqrt{49}\right)-\left(\left(3\right)\sqrt{28}\right)+\dfrac{20}{3}+\dfrac{56}{3}-\left(\left(\dfrac{47}{4}\right)\sqrt{175}\right)-\dfrac{25}{3}+\left(9\right)\sqrt{63}+\left(\dfrac{16}{3}\right)\sqrt{28}+\left(\dfrac{29}{2}\right)\sqrt{28}+\left(-\dfrac{31}{7}\right)\sqrt{63}+\left(-\dfrac{7}{8}\right)\sqrt{28}\) . 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{25}{8}\right)\sqrt{49}\right)+\left(\left(\left(\dfrac{16}{3}\right)\sqrt{28}\right)-\left(\left(-7\right)\sqrt{49}\right)-\left(\left(3\right)\sqrt{28}\right)+\dfrac{20}{3}+\dfrac{56}{3}-\left(\left(\dfrac{47}{4}\right)\sqrt{175}\right)-\dfrac{25}{3}+\left(9\right)\sqrt{63}+\left(\dfrac{16}{3}\right)\sqrt{28}+\left(\dfrac{29}{2}\right)\sqrt{28}+\left(-\dfrac{31}{7}\right)\sqrt{63}+\left(-\dfrac{7}{8}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{175}{8}\right)+\left(\left(\left(\dfrac{32}{3}\right)\sqrt{7}\right)+49-\left(\left(6\right)\sqrt{7}\right)+\dfrac{20}{3}+\dfrac{56}{3}-\left(\left(\dfrac{235}{4}\right)\sqrt{7}\right)-\dfrac{25}{3}+\left(27\right)\sqrt{7}+\left(\dfrac{32}{3}\right)\sqrt{7}+\left(29\right)\sqrt{7}+\left(-\dfrac{93}{7}\right)\sqrt{7}+\left(-\dfrac{7}{4}\right)\sqrt{7}\right)\\
&=&-\dfrac{175}{8}+\left(\left(\dfrac{32}{3}\right)\sqrt{7}\right)+49-\left(\left(6\right)\sqrt{7}\right)+\dfrac{20}{3}+\dfrac{56}{3}-\left(\left(\dfrac{235}{4}\right)\sqrt{7}\right)-\dfrac{25}{3}+\left(27\right)\sqrt{7}+\left(\dfrac{32}{3}\right)\sqrt{7}+\left(29\right)\sqrt{7}+\left(-\dfrac{93}{7}\right)\sqrt{7}+\left(-\dfrac{7}{4}\right)\sqrt{7}\\
&=&\dfrac{353}{8}+\left(-\dfrac{103}{42}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{25}{8}\right)\sqrt{49}\right)-\left(\left(\left(\dfrac{16}{3}\right)\sqrt{28}\right)-\left(\left(-7\right)\sqrt{49}\right)-\left(\left(3\right)\sqrt{28}\right)+\dfrac{20}{3}+\dfrac{56}{3}-\left(\left(\dfrac{47}{4}\right)\sqrt{175}\right)-\dfrac{25}{3}+\left(9\right)\sqrt{63}+\left(\dfrac{16}{3}\right)\sqrt{28}+\left(\dfrac{29}{2}\right)\sqrt{28}+\left(-\dfrac{31}{7}\right)\sqrt{63}+\left(-\dfrac{7}{8}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{175}{8}\right)-\left(\left(\left(\dfrac{32}{3}\right)\sqrt{7}\right)+49-\left(\left(6\right)\sqrt{7}\right)+\dfrac{20}{3}+\dfrac{56}{3}-\left(\left(\dfrac{235}{4}\right)\sqrt{7}\right)-\dfrac{25}{3}+\left(27\right)\sqrt{7}+\left(\dfrac{32}{3}\right)\sqrt{7}+\left(29\right)\sqrt{7}+\left(-\dfrac{93}{7}\right)\sqrt{7}+\left(-\dfrac{7}{4}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{175}{8}\right)-\left(\left(-\dfrac{103}{42}\right)\sqrt{7}+66\right)\\
&=&-\dfrac{175}{8}+\left(\dfrac{103}{42}\right)\sqrt{7}-66\\
&=&-\dfrac{703}{8}+\left(\dfrac{103}{42}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{25}{8}\right)\sqrt{49}\right)\times\left(\left(\left(\dfrac{16}{3}\right)\sqrt{28}\right)-\left(\left(-7\right)\sqrt{49}\right)-\left(\left(3\right)\sqrt{28}\right)+\dfrac{20}{3}+\dfrac{56}{3}-\left(\left(\dfrac{47}{4}\right)\sqrt{175}\right)-\dfrac{25}{3}+\left(9\right)\sqrt{63}+\left(\dfrac{16}{3}\right)\sqrt{28}+\left(\dfrac{29}{2}\right)\sqrt{28}+\left(-\dfrac{31}{7}\right)\sqrt{63}+\left(-\dfrac{7}{8}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{175}{8}\right)\times\left(\left(\left(\dfrac{32}{3}\right)\sqrt{7}\right)+49-\left(\left(6\right)\sqrt{7}\right)+\dfrac{20}{3}+\dfrac{56}{3}-\left(\left(\dfrac{235}{4}\right)\sqrt{7}\right)-\dfrac{25}{3}+\left(27\right)\sqrt{7}+\left(\dfrac{32}{3}\right)\sqrt{7}+\left(29\right)\sqrt{7}+\left(-\dfrac{93}{7}\right)\sqrt{7}+\left(-\dfrac{7}{4}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{175}{8}\right)\left(\left(-\dfrac{103}{42}\right)\sqrt{7}+66\right)\\
&=&\left(\dfrac{2575}{48}\right)\sqrt{7}-\dfrac{5775}{4}\\
&=&\left(\dfrac{2575}{48}\right)\sqrt{7}-\dfrac{5775}{4}\\
\end{eqnarray*}