L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(9\right)\sqrt{8}+\left(-\dfrac{40}{7}\right)\sqrt{18}-8+\left(-6\right)\sqrt{18}+5+\left(-\dfrac{32}{9}\right)\sqrt{8}+\left(-\dfrac{79}{9}\right)\sqrt{8}\) et \( Y=\left(\left(-\dfrac{67}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{18}\right)-\left(\left(-1\right)\sqrt{8}\right)+\dfrac{67}{4}+\left(-3\right)\sqrt{18}+\left(-3\right)\sqrt{18}+\left(\dfrac{11}{3}\right)\sqrt{50}+0+\left(-\dfrac{63}{4}\right)\sqrt{4}+\left(\dfrac{22}{3}\right)\sqrt{18}+\left(-3\right)\sqrt{4}+\left(-\dfrac{16}{3}\right)\sqrt{18}+\left(\dfrac{25}{7}\right)\sqrt{18}+\left(6\right)\sqrt{8}+\left(-\dfrac{67}{3}\right)\sqrt{18}\) . 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{8}+\left(-\dfrac{40}{7}\right)\sqrt{18}-8+\left(-6\right)\sqrt{18}+5+\left(-\dfrac{32}{9}\right)\sqrt{8}+\left(-\dfrac{79}{9}\right)\sqrt{8}\right)+\left(\left(\left(-\dfrac{67}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{18}\right)-\left(\left(-1\right)\sqrt{8}\right)+\dfrac{67}{4}+\left(-3\right)\sqrt{18}+\left(-3\right)\sqrt{18}+\left(\dfrac{11}{3}\right)\sqrt{50}+0+\left(-\dfrac{63}{4}\right)\sqrt{4}+\left(\dfrac{22}{3}\right)\sqrt{18}+\left(-3\right)\sqrt{4}+\left(-\dfrac{16}{3}\right)\sqrt{18}+\left(\dfrac{25}{7}\right)\sqrt{18}+\left(6\right)\sqrt{8}+\left(-\dfrac{67}{3}\right)\sqrt{18}\right)\\
&=&\left(\left(18\right)\sqrt{2}+\left(-\dfrac{120}{7}\right)\sqrt{2}-8+\left(-18\right)\sqrt{2}+5+\left(-\dfrac{64}{9}\right)\sqrt{2}+\left(-\dfrac{158}{9}\right)\sqrt{2}\right)+\left(\left(\left(-\dfrac{201}{8}\right)\sqrt{2}\right)-\left(\left(-67\right)\sqrt{2}\right)-\left(\left(-2\right)\sqrt{2}\right)+\dfrac{67}{4}+\left(-9\right)\sqrt{2}+\left(-9\right)\sqrt{2}+\left(\dfrac{55}{3}\right)\sqrt{2}+0-\dfrac{63}{2}+\left(22\right)\sqrt{2}-6+\left(-16\right)\sqrt{2}+\left(\dfrac{75}{7}\right)\sqrt{2}+\left(12\right)\sqrt{2}+\left(-67\right)\sqrt{2}\right)\\
&=&\left(18\right)\sqrt{2}+\left(-\dfrac{120}{7}\right)\sqrt{2}-8+\left(-18\right)\sqrt{2}+5+\left(-\dfrac{64}{9}\right)\sqrt{2}+\left(-\dfrac{158}{9}\right)\sqrt{2}+\left(\left(-\dfrac{201}{8}\right)\sqrt{2}\right)-\left(\left(-67\right)\sqrt{2}\right)-\left(\left(-2\right)\sqrt{2}\right)+\dfrac{67}{4}+\left(-9\right)\sqrt{2}+\left(-9\right)\sqrt{2}+\left(\dfrac{55}{3}\right)\sqrt{2}+0-\dfrac{63}{2}+\left(22\right)\sqrt{2}-6+\left(-16\right)\sqrt{2}+\left(\dfrac{75}{7}\right)\sqrt{2}+\left(12\right)\sqrt{2}+\left(-67\right)\sqrt{2}\\
&=&\left(-\dfrac{6029}{168}\right)\sqrt{2}-\dfrac{95}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(9\right)\sqrt{8}+\left(-\dfrac{40}{7}\right)\sqrt{18}-8+\left(-6\right)\sqrt{18}+5+\left(-\dfrac{32}{9}\right)\sqrt{8}+\left(-\dfrac{79}{9}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{67}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{18}\right)-\left(\left(-1\right)\sqrt{8}\right)+\dfrac{67}{4}+\left(-3\right)\sqrt{18}+\left(-3\right)\sqrt{18}+\left(\dfrac{11}{3}\right)\sqrt{50}+0+\left(-\dfrac{63}{4}\right)\sqrt{4}+\left(\dfrac{22}{3}\right)\sqrt{18}+\left(-3\right)\sqrt{4}+\left(-\dfrac{16}{3}\right)\sqrt{18}+\left(\dfrac{25}{7}\right)\sqrt{18}+\left(6\right)\sqrt{8}+\left(-\dfrac{67}{3}\right)\sqrt{18}\right)\\
&=&\left(\left(18\right)\sqrt{2}+\left(-\dfrac{120}{7}\right)\sqrt{2}-8+\left(-18\right)\sqrt{2}+5+\left(-\dfrac{64}{9}\right)\sqrt{2}+\left(-\dfrac{158}{9}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{201}{8}\right)\sqrt{2}\right)-\left(\left(-67\right)\sqrt{2}\right)-\left(\left(-2\right)\sqrt{2}\right)+\dfrac{67}{4}+\left(-9\right)\sqrt{2}+\left(-9\right)\sqrt{2}+\left(\dfrac{55}{3}\right)\sqrt{2}+0-\dfrac{63}{2}+\left(22\right)\sqrt{2}-6+\left(-16\right)\sqrt{2}+\left(\dfrac{75}{7}\right)\sqrt{2}+\left(12\right)\sqrt{2}+\left(-67\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{878}{21}\right)\sqrt{2}-3\right)-\left(\left(\dfrac{995}{168}\right)\sqrt{2}-\dfrac{83}{4}\right)\\
&=&\left(-\dfrac{878}{21}\right)\sqrt{2}-3+\left(-\dfrac{995}{168}\right)\sqrt{2}+\dfrac{83}{4}\\
&=&\left(-\dfrac{2673}{56}\right)\sqrt{2}+\dfrac{71}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(9\right)\sqrt{8}+\left(-\dfrac{40}{7}\right)\sqrt{18}-8+\left(-6\right)\sqrt{18}+5+\left(-\dfrac{32}{9}\right)\sqrt{8}+\left(-\dfrac{79}{9}\right)\sqrt{8}\right)\times\left(\left(\left(-\dfrac{67}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{18}\right)-\left(\left(-1\right)\sqrt{8}\right)+\dfrac{67}{4}+\left(-3\right)\sqrt{18}+\left(-3\right)\sqrt{18}+\left(\dfrac{11}{3}\right)\sqrt{50}+0+\left(-\dfrac{63}{4}\right)\sqrt{4}+\left(\dfrac{22}{3}\right)\sqrt{18}+\left(-3\right)\sqrt{4}+\left(-\dfrac{16}{3}\right)\sqrt{18}+\left(\dfrac{25}{7}\right)\sqrt{18}+\left(6\right)\sqrt{8}+\left(-\dfrac{67}{3}\right)\sqrt{18}\right)\\
&=&\left(\left(18\right)\sqrt{2}+\left(-\dfrac{120}{7}\right)\sqrt{2}-8+\left(-18\right)\sqrt{2}+5+\left(-\dfrac{64}{9}\right)\sqrt{2}+\left(-\dfrac{158}{9}\right)\sqrt{2}\right)\times\left(\left(\left(-\dfrac{201}{8}\right)\sqrt{2}\right)-\left(\left(-67\right)\sqrt{2}\right)-\left(\left(-2\right)\sqrt{2}\right)+\dfrac{67}{4}+\left(-9\right)\sqrt{2}+\left(-9\right)\sqrt{2}+\left(\dfrac{55}{3}\right)\sqrt{2}+0-\dfrac{63}{2}+\left(22\right)\sqrt{2}-6+\left(-16\right)\sqrt{2}+\left(\dfrac{75}{7}\right)\sqrt{2}+\left(12\right)\sqrt{2}+\left(-67\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{878}{21}\right)\sqrt{2}-3\right)\left(\left(\dfrac{995}{168}\right)\sqrt{2}-\dfrac{83}{4}\right)\\
&=&\left(-\dfrac{436805}{1764}\right)\sqrt{4}+\left(\dfrac{142763}{168}\right)\sqrt{2}+\dfrac{249}{4}\\
&=&-\dfrac{763801}{1764}+\left(\dfrac{142763}{168}\right)\sqrt{2}\\
\end{eqnarray*}