L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(9\right)\sqrt{4}+\left(-5\right)\sqrt{8}+\left(-\dfrac{17}{3}\right)\sqrt{50}-\dfrac{1}{3}+\left(-1\right)\sqrt{18}+\left(\dfrac{78}{5}\right)\sqrt{8}-\dfrac{50}{3}+\left(-\dfrac{17}{7}\right)\sqrt{18}-\dfrac{47}{8}+\left(-\dfrac{70}{9}\right)\sqrt{4}+\left(\dfrac{77}{4}\right)\sqrt{4}+\left(\dfrac{73}{8}\right)\sqrt{4}\) et \( Y=\left(-\dfrac{22}{3}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(9\right)\sqrt{4}+\left(-5\right)\sqrt{8}+\left(-\dfrac{17}{3}\right)\sqrt{50}-\dfrac{1}{3}+\left(-1\right)\sqrt{18}+\left(\dfrac{78}{5}\right)\sqrt{8}-\dfrac{50}{3}+\left(-\dfrac{17}{7}\right)\sqrt{18}-\dfrac{47}{8}+\left(-\dfrac{70}{9}\right)\sqrt{4}+\left(\dfrac{77}{4}\right)\sqrt{4}+\left(\dfrac{73}{8}\right)\sqrt{4}\right)+\left(\left(-\dfrac{22}{3}\right)\sqrt{18}\right)\\
&=&\left(18+\left(-10\right)\sqrt{2}+\left(-\dfrac{85}{3}\right)\sqrt{2}-\dfrac{1}{3}+\left(-3\right)\sqrt{2}+\left(\dfrac{156}{5}\right)\sqrt{2}-\dfrac{50}{3}+\left(-\dfrac{51}{7}\right)\sqrt{2}-\dfrac{47}{8}-\dfrac{140}{9}+\dfrac{77}{2}+\dfrac{73}{4}\right)+\left(\left(-22\right)\sqrt{2}\right)\\
&=&18+\left(-10\right)\sqrt{2}+\left(-\dfrac{85}{3}\right)\sqrt{2}-\dfrac{1}{3}+\left(-3\right)\sqrt{2}+\left(\dfrac{156}{5}\right)\sqrt{2}-\dfrac{50}{3}+\left(-\dfrac{51}{7}\right)\sqrt{2}-\dfrac{47}{8}-\dfrac{140}{9}+\dfrac{77}{2}+\dfrac{73}{4}+\left(-22\right)\sqrt{2}\\
&=&\dfrac{2615}{72}+\left(-\dfrac{4139}{105}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(9\right)\sqrt{4}+\left(-5\right)\sqrt{8}+\left(-\dfrac{17}{3}\right)\sqrt{50}-\dfrac{1}{3}+\left(-1\right)\sqrt{18}+\left(\dfrac{78}{5}\right)\sqrt{8}-\dfrac{50}{3}+\left(-\dfrac{17}{7}\right)\sqrt{18}-\dfrac{47}{8}+\left(-\dfrac{70}{9}\right)\sqrt{4}+\left(\dfrac{77}{4}\right)\sqrt{4}+\left(\dfrac{73}{8}\right)\sqrt{4}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{18}\right)\\
&=&\left(18+\left(-10\right)\sqrt{2}+\left(-\dfrac{85}{3}\right)\sqrt{2}-\dfrac{1}{3}+\left(-3\right)\sqrt{2}+\left(\dfrac{156}{5}\right)\sqrt{2}-\dfrac{50}{3}+\left(-\dfrac{51}{7}\right)\sqrt{2}-\dfrac{47}{8}-\dfrac{140}{9}+\dfrac{77}{2}+\dfrac{73}{4}\right)-\left(\left(-22\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2615}{72}+\left(-\dfrac{1829}{105}\right)\sqrt{2}\right)-\left(\left(-22\right)\sqrt{2}\right)\\
&=&\dfrac{2615}{72}+\left(-\dfrac{1829}{105}\right)\sqrt{2}+\left(22\right)\sqrt{2}\\
&=&\dfrac{2615}{72}+\left(\dfrac{481}{105}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(9\right)\sqrt{4}+\left(-5\right)\sqrt{8}+\left(-\dfrac{17}{3}\right)\sqrt{50}-\dfrac{1}{3}+\left(-1\right)\sqrt{18}+\left(\dfrac{78}{5}\right)\sqrt{8}-\dfrac{50}{3}+\left(-\dfrac{17}{7}\right)\sqrt{18}-\dfrac{47}{8}+\left(-\dfrac{70}{9}\right)\sqrt{4}+\left(\dfrac{77}{4}\right)\sqrt{4}+\left(\dfrac{73}{8}\right)\sqrt{4}\right)\times\left(\left(-\dfrac{22}{3}\right)\sqrt{18}\right)\\
&=&\left(18+\left(-10\right)\sqrt{2}+\left(-\dfrac{85}{3}\right)\sqrt{2}-\dfrac{1}{3}+\left(-3\right)\sqrt{2}+\left(\dfrac{156}{5}\right)\sqrt{2}-\dfrac{50}{3}+\left(-\dfrac{51}{7}\right)\sqrt{2}-\dfrac{47}{8}-\dfrac{140}{9}+\dfrac{77}{2}+\dfrac{73}{4}\right)\times\left(\left(-22\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2615}{72}+\left(-\dfrac{1829}{105}\right)\sqrt{2}\right)\left(\left(-22\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{28765}{36}\right)\sqrt{2}+\left(\dfrac{40238}{105}\right)\sqrt{4}\\
&=&\left(-\dfrac{28765}{36}\right)\sqrt{2}+\dfrac{80476}{105}\\
\end{eqnarray*}