L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(6\right)\sqrt{18}\right)+\dfrac{66}{5}-\left(\left(-\dfrac{59}{6}\right)\sqrt{8}\right)-\left(\left(-\dfrac{32}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{78}{7}\right)\sqrt{18}\right)+\left(0\right)\sqrt{50}+\left(-\dfrac{17}{9}\right)\sqrt{50}+\dfrac{15}{4}+\dfrac{11}{9}-7+\left(6\right)\sqrt{50}+\left(8\right)\sqrt{50}+\left(\dfrac{13}{2}\right)\sqrt{50}+\left(\dfrac{59}{4}\right)\sqrt{50}+\left(-4\right)\sqrt{50}+\left(0\right)\sqrt{18}\) et \( Y=\left(-\dfrac{11}{8}\right)\sqrt{4}+\left(-\dfrac{81}{4}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(6\right)\sqrt{18}\right)+\dfrac{66}{5}-\left(\left(-\dfrac{59}{6}\right)\sqrt{8}\right)-\left(\left(-\dfrac{32}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{78}{7}\right)\sqrt{18}\right)+\left(0\right)\sqrt{50}+\left(-\dfrac{17}{9}\right)\sqrt{50}+\dfrac{15}{4}+\dfrac{11}{9}-7+\left(6\right)\sqrt{50}+\left(8\right)\sqrt{50}+\left(\dfrac{13}{2}\right)\sqrt{50}+\left(\dfrac{59}{4}\right)\sqrt{50}+\left(-4\right)\sqrt{50}+\left(0\right)\sqrt{18}\right)+\left(\left(-\dfrac{11}{8}\right)\sqrt{4}+\left(-\dfrac{81}{4}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(18\right)\sqrt{2}\right)+\dfrac{66}{5}-\left(\left(-\dfrac{59}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{160}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{234}{7}\right)\sqrt{2}\right)+\left(0\right)\sqrt{2}+\left(-\dfrac{85}{9}\right)\sqrt{2}+\dfrac{15}{4}+\dfrac{11}{9}-7+\left(30\right)\sqrt{2}+\left(40\right)\sqrt{2}+\left(\dfrac{65}{2}\right)\sqrt{2}+\left(\dfrac{295}{4}\right)\sqrt{2}+\left(-20\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)+\left(-\dfrac{11}{4}+\left(-\dfrac{243}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(18\right)\sqrt{2}\right)+\dfrac{66}{5}-\left(\left(-\dfrac{59}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{160}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{234}{7}\right)\sqrt{2}\right)+\left(0\right)\sqrt{2}+\left(-\dfrac{85}{9}\right)\sqrt{2}+\dfrac{15}{4}+\dfrac{11}{9}-7+\left(30\right)\sqrt{2}+\left(40\right)\sqrt{2}+\left(\dfrac{65}{2}\right)\sqrt{2}+\left(\dfrac{295}{4}\right)\sqrt{2}+\left(-20\right)\sqrt{2}+\left(0\right)\sqrt{2}-\dfrac{11}{4}+\left(-\dfrac{243}{4}\right)\sqrt{2}\\
&=&\left(\dfrac{2449}{14}\right)\sqrt{2}+\dfrac{379}{45}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(6\right)\sqrt{18}\right)+\dfrac{66}{5}-\left(\left(-\dfrac{59}{6}\right)\sqrt{8}\right)-\left(\left(-\dfrac{32}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{78}{7}\right)\sqrt{18}\right)+\left(0\right)\sqrt{50}+\left(-\dfrac{17}{9}\right)\sqrt{50}+\dfrac{15}{4}+\dfrac{11}{9}-7+\left(6\right)\sqrt{50}+\left(8\right)\sqrt{50}+\left(\dfrac{13}{2}\right)\sqrt{50}+\left(\dfrac{59}{4}\right)\sqrt{50}+\left(-4\right)\sqrt{50}+\left(0\right)\sqrt{18}\right)-\left(\left(-\dfrac{11}{8}\right)\sqrt{4}+\left(-\dfrac{81}{4}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(18\right)\sqrt{2}\right)+\dfrac{66}{5}-\left(\left(-\dfrac{59}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{160}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{234}{7}\right)\sqrt{2}\right)+\left(0\right)\sqrt{2}+\left(-\dfrac{85}{9}\right)\sqrt{2}+\dfrac{15}{4}+\dfrac{11}{9}-7+\left(30\right)\sqrt{2}+\left(40\right)\sqrt{2}+\left(\dfrac{65}{2}\right)\sqrt{2}+\left(\dfrac{295}{4}\right)\sqrt{2}+\left(-20\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)-\left(-\dfrac{11}{4}+\left(-\dfrac{243}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{6599}{28}\right)\sqrt{2}+\dfrac{2011}{180}\right)-\left(-\dfrac{11}{4}+\left(-\dfrac{243}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{6599}{28}\right)\sqrt{2}+\dfrac{2011}{180}+\dfrac{11}{4}+\left(\dfrac{243}{4}\right)\sqrt{2}\\
&=&\left(\dfrac{2075}{7}\right)\sqrt{2}+\dfrac{1253}{90}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(6\right)\sqrt{18}\right)+\dfrac{66}{5}-\left(\left(-\dfrac{59}{6}\right)\sqrt{8}\right)-\left(\left(-\dfrac{32}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{78}{7}\right)\sqrt{18}\right)+\left(0\right)\sqrt{50}+\left(-\dfrac{17}{9}\right)\sqrt{50}+\dfrac{15}{4}+\dfrac{11}{9}-7+\left(6\right)\sqrt{50}+\left(8\right)\sqrt{50}+\left(\dfrac{13}{2}\right)\sqrt{50}+\left(\dfrac{59}{4}\right)\sqrt{50}+\left(-4\right)\sqrt{50}+\left(0\right)\sqrt{18}\right)\times\left(\left(-\dfrac{11}{8}\right)\sqrt{4}+\left(-\dfrac{81}{4}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(18\right)\sqrt{2}\right)+\dfrac{66}{5}-\left(\left(-\dfrac{59}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{160}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{234}{7}\right)\sqrt{2}\right)+\left(0\right)\sqrt{2}+\left(-\dfrac{85}{9}\right)\sqrt{2}+\dfrac{15}{4}+\dfrac{11}{9}-7+\left(30\right)\sqrt{2}+\left(40\right)\sqrt{2}+\left(\dfrac{65}{2}\right)\sqrt{2}+\left(\dfrac{295}{4}\right)\sqrt{2}+\left(-20\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\times\left(-\dfrac{11}{4}+\left(-\dfrac{243}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{6599}{28}\right)\sqrt{2}+\dfrac{2011}{180}\right)\left(-\dfrac{11}{4}+\left(-\dfrac{243}{4}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{46439}{35}\right)\sqrt{2}+\left(-\dfrac{1603557}{112}\right)\sqrt{4}-\dfrac{22121}{720}\\
&=&\left(-\dfrac{46439}{35}\right)\sqrt{2}-\dfrac{144474977}{5040}\\
\end{eqnarray*}