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 \( X=\left(\left(-\dfrac{5}{3}\right)\sqrt{18}+\left(-\dfrac{48}{5}\right)\sqrt{18}+\left(4\right)\sqrt{18}\right)-\left(\left(\dfrac{60}{7}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(\dfrac{7}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{8}+\left(\dfrac{47}{3}\right)\sqrt{4}+\left(\dfrac{59}{7}\right)\sqrt{8}+\left(7\right)\sqrt{8}\right)-\left(\left(\left(-9\right)\sqrt{50}\right)-\left(\left(4\right)\sqrt{18}\right)\right)\) et \( Y=\left(-1\right)\sqrt{50}\) . 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{5}{3}\right)\sqrt{18}+\left(-\dfrac{48}{5}\right)\sqrt{18}+\left(4\right)\sqrt{18}\right)-\left(\left(\dfrac{60}{7}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(\dfrac{7}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{8}+\left(\dfrac{47}{3}\right)\sqrt{4}+\left(\dfrac{59}{7}\right)\sqrt{8}+\left(7\right)\sqrt{8}\right)-\left(\left(\left(-9\right)\sqrt{50}\right)-\left(\left(4\right)\sqrt{18}\right)\right)\right)+\left(\left(-1\right)\sqrt{50}\right)\\
&=&\left(\left(\left(-5\right)\sqrt{2}+\left(-\dfrac{144}{5}\right)\sqrt{2}+\left(12\right)\sqrt{2}\right)-\left(\dfrac{120}{7}+\left(5\right)\sqrt{2}+\left(\dfrac{14}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{2}+\dfrac{94}{3}+\left(\dfrac{118}{7}\right)\sqrt{2}+\left(14\right)\sqrt{2}\right)-\left(\left(\left(-45\right)\sqrt{2}\right)-\left(\left(12\right)\sqrt{2}\right)\right)\right)+\left(\left(-5\right)\sqrt{2}\right)\\
&=&\left(\left(-5\right)\sqrt{2}+\left(-\dfrac{144}{5}\right)\sqrt{2}+\left(12\right)\sqrt{2}\right)-\left(\dfrac{120}{7}+\left(5\right)\sqrt{2}+\left(\dfrac{14}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{2}+\dfrac{94}{3}+\left(\dfrac{118}{7}\right)\sqrt{2}+\left(14\right)\sqrt{2}\right)-\left(\left(\left(-45\right)\sqrt{2}\right)-\left(\left(12\right)\sqrt{2}\right)\right)+\left(-5\right)\sqrt{2}\\
&=&\left(-\dfrac{1852}{315}\right)\sqrt{2}-\dfrac{1018}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{5}{3}\right)\sqrt{18}+\left(-\dfrac{48}{5}\right)\sqrt{18}+\left(4\right)\sqrt{18}\right)-\left(\left(\dfrac{60}{7}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(\dfrac{7}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{8}+\left(\dfrac{47}{3}\right)\sqrt{4}+\left(\dfrac{59}{7}\right)\sqrt{8}+\left(7\right)\sqrt{8}\right)-\left(\left(\left(-9\right)\sqrt{50}\right)-\left(\left(4\right)\sqrt{18}\right)\right)\right)-\left(\left(-1\right)\sqrt{50}\right)\\
&=&\left(\left(\left(-5\right)\sqrt{2}+\left(-\dfrac{144}{5}\right)\sqrt{2}+\left(12\right)\sqrt{2}\right)-\left(\dfrac{120}{7}+\left(5\right)\sqrt{2}+\left(\dfrac{14}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{2}+\dfrac{94}{3}+\left(\dfrac{118}{7}\right)\sqrt{2}+\left(14\right)\sqrt{2}\right)-\left(\left(\left(-45\right)\sqrt{2}\right)-\left(\left(12\right)\sqrt{2}\right)\right)\right)-\left(\left(-5\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{277}{315}\right)\sqrt{2}-\dfrac{1018}{21}\right)-\left(\left(-5\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{277}{315}\right)\sqrt{2}-\dfrac{1018}{21}+\left(5\right)\sqrt{2}\\
&=&\left(\dfrac{1298}{315}\right)\sqrt{2}-\dfrac{1018}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{5}{3}\right)\sqrt{18}+\left(-\dfrac{48}{5}\right)\sqrt{18}+\left(4\right)\sqrt{18}\right)-\left(\left(\dfrac{60}{7}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(\dfrac{7}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{8}+\left(\dfrac{47}{3}\right)\sqrt{4}+\left(\dfrac{59}{7}\right)\sqrt{8}+\left(7\right)\sqrt{8}\right)-\left(\left(\left(-9\right)\sqrt{50}\right)-\left(\left(4\right)\sqrt{18}\right)\right)\right)\times\left(\left(-1\right)\sqrt{50}\right)\\
&=&\left(\left(\left(-5\right)\sqrt{2}+\left(-\dfrac{144}{5}\right)\sqrt{2}+\left(12\right)\sqrt{2}\right)-\left(\dfrac{120}{7}+\left(5\right)\sqrt{2}+\left(\dfrac{14}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{2}+\dfrac{94}{3}+\left(\dfrac{118}{7}\right)\sqrt{2}+\left(14\right)\sqrt{2}\right)-\left(\left(\left(-45\right)\sqrt{2}\right)-\left(\left(12\right)\sqrt{2}\right)\right)\right)\times\left(\left(-5\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{277}{315}\right)\sqrt{2}-\dfrac{1018}{21}\right)\left(\left(-5\right)\sqrt{2}\right)\\
&=&\left(\dfrac{277}{63}\right)\sqrt{4}+\left(\dfrac{5090}{21}\right)\sqrt{2}\\
&=&\dfrac{554}{63}+\left(\dfrac{5090}{21}\right)\sqrt{2}\\
\end{eqnarray*}