L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(8\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{1}{6}\right)\sqrt{8}\right)-\dfrac{59}{3}+\left(\left(-\dfrac{5}{7}\right)\sqrt{18}\right)-\left(\left(-8\right)\sqrt{8}\right)-\left(\left(-2\right)\sqrt{8}\right)-\left(\left(-5\right)\sqrt{50}\right)-\left(\left(-\dfrac{5}{7}\right)\sqrt{18}\right)+\left(\left(-\dfrac{63}{5}\right)\sqrt{4}\right)-\left(\left(-\dfrac{32}{5}\right)\sqrt{4}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{50}\right)\) et \( Y=\left(-3\right)\sqrt{8}\) . 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(8\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{1}{6}\right)\sqrt{8}\right)-\dfrac{59}{3}+\left(\left(-\dfrac{5}{7}\right)\sqrt{18}\right)-\left(\left(-8\right)\sqrt{8}\right)-\left(\left(-2\right)\sqrt{8}\right)-\left(\left(-5\right)\sqrt{50}\right)-\left(\left(-\dfrac{5}{7}\right)\sqrt{18}\right)+\left(\left(-\dfrac{63}{5}\right)\sqrt{4}\right)-\left(\left(-\dfrac{32}{5}\right)\sqrt{4}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{50}\right)\right)+\left(\left(-3\right)\sqrt{8}\right)\\
&=&\left(\left(\left(24\right)\sqrt{2}\right)-\left(\left(\dfrac{57}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{1}{3}\right)\sqrt{2}\right)-\dfrac{59}{3}+\left(\left(-\dfrac{15}{7}\right)\sqrt{2}\right)-\left(\left(-16\right)\sqrt{2}\right)-\left(\left(-4\right)\sqrt{2}\right)-\left(\left(-25\right)\sqrt{2}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{2}\right)-\dfrac{126}{5}+\dfrac{64}{5}-\left(\left(\dfrac{180}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{180}{7}\right)\sqrt{2}\right)-\left(\left(-15\right)\sqrt{2}\right)\right)+\left(\left(-6\right)\sqrt{2}\right)\\
&=&\left(\left(24\right)\sqrt{2}\right)-\left(\left(\dfrac{57}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{1}{3}\right)\sqrt{2}\right)-\dfrac{59}{3}+\left(\left(-\dfrac{15}{7}\right)\sqrt{2}\right)-\left(\left(-16\right)\sqrt{2}\right)-\left(\left(-4\right)\sqrt{2}\right)-\left(\left(-25\right)\sqrt{2}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{2}\right)-\dfrac{126}{5}+\dfrac{64}{5}-\left(\left(\dfrac{180}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{180}{7}\right)\sqrt{2}\right)-\left(\left(-15\right)\sqrt{2}\right)+\left(-6\right)\sqrt{2}\\
&=&\left(\dfrac{1558}{105}\right)\sqrt{2}-\dfrac{481}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(8\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{1}{6}\right)\sqrt{8}\right)-\dfrac{59}{3}+\left(\left(-\dfrac{5}{7}\right)\sqrt{18}\right)-\left(\left(-8\right)\sqrt{8}\right)-\left(\left(-2\right)\sqrt{8}\right)-\left(\left(-5\right)\sqrt{50}\right)-\left(\left(-\dfrac{5}{7}\right)\sqrt{18}\right)+\left(\left(-\dfrac{63}{5}\right)\sqrt{4}\right)-\left(\left(-\dfrac{32}{5}\right)\sqrt{4}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{50}\right)\right)-\left(\left(-3\right)\sqrt{8}\right)\\
&=&\left(\left(\left(24\right)\sqrt{2}\right)-\left(\left(\dfrac{57}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{1}{3}\right)\sqrt{2}\right)-\dfrac{59}{3}+\left(\left(-\dfrac{15}{7}\right)\sqrt{2}\right)-\left(\left(-16\right)\sqrt{2}\right)-\left(\left(-4\right)\sqrt{2}\right)-\left(\left(-25\right)\sqrt{2}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{2}\right)-\dfrac{126}{5}+\dfrac{64}{5}-\left(\left(\dfrac{180}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{180}{7}\right)\sqrt{2}\right)-\left(\left(-15\right)\sqrt{2}\right)\right)-\left(\left(-6\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{2188}{105}\right)\sqrt{2}-\dfrac{481}{15}\right)-\left(\left(-6\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2188}{105}\right)\sqrt{2}-\dfrac{481}{15}+\left(6\right)\sqrt{2}\\
&=&\left(\dfrac{2818}{105}\right)\sqrt{2}-\dfrac{481}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(8\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{1}{6}\right)\sqrt{8}\right)-\dfrac{59}{3}+\left(\left(-\dfrac{5}{7}\right)\sqrt{18}\right)-\left(\left(-8\right)\sqrt{8}\right)-\left(\left(-2\right)\sqrt{8}\right)-\left(\left(-5\right)\sqrt{50}\right)-\left(\left(-\dfrac{5}{7}\right)\sqrt{18}\right)+\left(\left(-\dfrac{63}{5}\right)\sqrt{4}\right)-\left(\left(-\dfrac{32}{5}\right)\sqrt{4}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{36}{7}\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{50}\right)\right)\times\left(\left(-3\right)\sqrt{8}\right)\\
&=&\left(\left(\left(24\right)\sqrt{2}\right)-\left(\left(\dfrac{57}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{1}{3}\right)\sqrt{2}\right)-\dfrac{59}{3}+\left(\left(-\dfrac{15}{7}\right)\sqrt{2}\right)-\left(\left(-16\right)\sqrt{2}\right)-\left(\left(-4\right)\sqrt{2}\right)-\left(\left(-25\right)\sqrt{2}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{2}\right)-\dfrac{126}{5}+\dfrac{64}{5}-\left(\left(\dfrac{180}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{180}{7}\right)\sqrt{2}\right)-\left(\left(-15\right)\sqrt{2}\right)\right)\times\left(\left(-6\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{2188}{105}\right)\sqrt{2}-\dfrac{481}{15}\right)\left(\left(-6\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{4376}{35}\right)\sqrt{4}+\left(\dfrac{962}{5}\right)\sqrt{2}\\
&=&-\dfrac{8752}{35}+\left(\dfrac{962}{5}\right)\sqrt{2}\\
\end{eqnarray*}