L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-\dfrac{31}{7}\right)\sqrt{18}\right)-3-\left(\left(\dfrac{47}{5}\right)\sqrt{8}\right)\right)-\left(\left(0\right)\sqrt{18}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{70}{3}\right)\sqrt{50}\right)-0\right)\) et \( Y=\left(\dfrac{1}{3}\right)\sqrt{8}\) . 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(\left(-\dfrac{31}{7}\right)\sqrt{18}\right)-3-\left(\left(\dfrac{47}{5}\right)\sqrt{8}\right)\right)-\left(\left(0\right)\sqrt{18}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{70}{3}\right)\sqrt{50}\right)-0\right)\right)+\left(\left(\dfrac{1}{3}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\left(-\dfrac{93}{7}\right)\sqrt{2}\right)-3-\left(\left(\dfrac{94}{5}\right)\sqrt{2}\right)\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(-62\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{350}{3}\right)\sqrt{2}\right)-0\right)\right)+\left(\left(\dfrac{2}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(\left(-\dfrac{93}{7}\right)\sqrt{2}\right)-3-\left(\left(\dfrac{94}{5}\right)\sqrt{2}\right)\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(-62\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{350}{3}\right)\sqrt{2}\right)-0\right)+\left(\dfrac{2}{3}\right)\sqrt{2}\\
&=&\left(\dfrac{15461}{105}\right)\sqrt{2}-3\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{31}{7}\right)\sqrt{18}\right)-3-\left(\left(\dfrac{47}{5}\right)\sqrt{8}\right)\right)-\left(\left(0\right)\sqrt{18}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{70}{3}\right)\sqrt{50}\right)-0\right)\right)-\left(\left(\dfrac{1}{3}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\left(-\dfrac{93}{7}\right)\sqrt{2}\right)-3-\left(\left(\dfrac{94}{5}\right)\sqrt{2}\right)\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(-62\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{350}{3}\right)\sqrt{2}\right)-0\right)\right)-\left(\left(\dfrac{2}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{15391}{105}\right)\sqrt{2}-3\right)-\left(\left(\dfrac{2}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{15391}{105}\right)\sqrt{2}-3+\left(-\dfrac{2}{3}\right)\sqrt{2}\\
&=&\left(\dfrac{5107}{35}\right)\sqrt{2}-3\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{31}{7}\right)\sqrt{18}\right)-3-\left(\left(\dfrac{47}{5}\right)\sqrt{8}\right)\right)-\left(\left(0\right)\sqrt{18}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{70}{3}\right)\sqrt{50}\right)-0\right)\right)\times\left(\left(\dfrac{1}{3}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\left(-\dfrac{93}{7}\right)\sqrt{2}\right)-3-\left(\left(\dfrac{94}{5}\right)\sqrt{2}\right)\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(-62\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{350}{3}\right)\sqrt{2}\right)-0\right)\right)\times\left(\left(\dfrac{2}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{15391}{105}\right)\sqrt{2}-3\right)\left(\left(\dfrac{2}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{30782}{315}\right)\sqrt{4}+\left(-2\right)\sqrt{2}\\
&=&\dfrac{61564}{315}+\left(-2\right)\sqrt{2}\\
\end{eqnarray*}