L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{9}{5}\right)\sqrt{45}+\left(-1\right)\sqrt{125}+\dfrac{49}{6}+\left(0\right)\sqrt{25}+\left(-\dfrac{17}{2}\right)\sqrt{125}+\dfrac{26}{9}+\left(\dfrac{49}{8}\right)\sqrt{45}+\left(0\right)\sqrt{45}-\dfrac{9}{2}\) et \( Y=\left(\dfrac{8}{3}\right)\sqrt{45}+\left(-6\right)\sqrt{20}+\left(\dfrac{38}{7}\right)\sqrt{125}-\dfrac{33}{4}+\left(-\dfrac{55}{4}\right)\sqrt{25}+\left(-\dfrac{5}{4}\right)\sqrt{20}-\dfrac{77}{8}+\left(-\dfrac{23}{9}\right)\sqrt{45}+\dfrac{74}{7}-\left(\left(-\dfrac{61}{7}\right)\sqrt{125}\right)-\dfrac{74}{7}-\left(\left(-\dfrac{10}{3}\right)\sqrt{125}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{9}{5}\right)\sqrt{45}+\left(-1\right)\sqrt{125}+\dfrac{49}{6}+\left(0\right)\sqrt{25}+\left(-\dfrac{17}{2}\right)\sqrt{125}+\dfrac{26}{9}+\left(\dfrac{49}{8}\right)\sqrt{45}+\left(0\right)\sqrt{45}-\dfrac{9}{2}\right)+\left(\left(\dfrac{8}{3}\right)\sqrt{45}+\left(-6\right)\sqrt{20}+\left(\dfrac{38}{7}\right)\sqrt{125}-\dfrac{33}{4}+\left(-\dfrac{55}{4}\right)\sqrt{25}+\left(-\dfrac{5}{4}\right)\sqrt{20}-\dfrac{77}{8}+\left(-\dfrac{23}{9}\right)\sqrt{45}+\dfrac{74}{7}-\left(\left(-\dfrac{61}{7}\right)\sqrt{125}\right)-\dfrac{74}{7}-\left(\left(-\dfrac{10}{3}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\dfrac{27}{5}\right)\sqrt{5}+\left(-5\right)\sqrt{5}+\dfrac{49}{6}+0+\left(-\dfrac{85}{2}\right)\sqrt{5}+\dfrac{26}{9}+\left(\dfrac{147}{8}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{9}{2}\right)+\left(\left(8\right)\sqrt{5}+\left(-12\right)\sqrt{5}+\left(\dfrac{190}{7}\right)\sqrt{5}-\dfrac{33}{4}-\dfrac{275}{4}+\left(-\dfrac{5}{2}\right)\sqrt{5}-\dfrac{77}{8}+\left(-\dfrac{23}{3}\right)\sqrt{5}+\dfrac{74}{7}-\left(\left(-\dfrac{305}{7}\right)\sqrt{5}\right)-\dfrac{74}{7}-\left(\left(-\dfrac{50}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{27}{5}\right)\sqrt{5}+\left(-5\right)\sqrt{5}+\dfrac{49}{6}+0+\left(-\dfrac{85}{2}\right)\sqrt{5}+\dfrac{26}{9}+\left(\dfrac{147}{8}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{9}{2}+\left(8\right)\sqrt{5}+\left(-12\right)\sqrt{5}+\left(\dfrac{190}{7}\right)\sqrt{5}-\dfrac{33}{4}-\dfrac{275}{4}+\left(-\dfrac{5}{2}\right)\sqrt{5}-\dfrac{77}{8}+\left(-\dfrac{23}{3}\right)\sqrt{5}+\dfrac{74}{7}-\left(\left(-\dfrac{305}{7}\right)\sqrt{5}\right)-\dfrac{74}{7}-\left(\left(-\dfrac{50}{3}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{13857}{280}\right)\sqrt{5}-\dfrac{5765}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{9}{5}\right)\sqrt{45}+\left(-1\right)\sqrt{125}+\dfrac{49}{6}+\left(0\right)\sqrt{25}+\left(-\dfrac{17}{2}\right)\sqrt{125}+\dfrac{26}{9}+\left(\dfrac{49}{8}\right)\sqrt{45}+\left(0\right)\sqrt{45}-\dfrac{9}{2}\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{45}+\left(-6\right)\sqrt{20}+\left(\dfrac{38}{7}\right)\sqrt{125}-\dfrac{33}{4}+\left(-\dfrac{55}{4}\right)\sqrt{25}+\left(-\dfrac{5}{4}\right)\sqrt{20}-\dfrac{77}{8}+\left(-\dfrac{23}{9}\right)\sqrt{45}+\dfrac{74}{7}-\left(\left(-\dfrac{61}{7}\right)\sqrt{125}\right)-\dfrac{74}{7}-\left(\left(-\dfrac{10}{3}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\dfrac{27}{5}\right)\sqrt{5}+\left(-5\right)\sqrt{5}+\dfrac{49}{6}+0+\left(-\dfrac{85}{2}\right)\sqrt{5}+\dfrac{26}{9}+\left(\dfrac{147}{8}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{9}{2}\right)-\left(\left(8\right)\sqrt{5}+\left(-12\right)\sqrt{5}+\left(\dfrac{190}{7}\right)\sqrt{5}-\dfrac{33}{4}-\dfrac{275}{4}+\left(-\dfrac{5}{2}\right)\sqrt{5}-\dfrac{77}{8}+\left(-\dfrac{23}{3}\right)\sqrt{5}+\dfrac{74}{7}-\left(\left(-\dfrac{305}{7}\right)\sqrt{5}\right)-\dfrac{74}{7}-\left(\left(-\dfrac{50}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{949}{40}\right)\sqrt{5}+\dfrac{59}{9}\right)-\left(\left(\dfrac{1025}{14}\right)\sqrt{5}-\dfrac{693}{8}\right)\\
&=&\left(-\dfrac{949}{40}\right)\sqrt{5}+\dfrac{59}{9}+\left(-\dfrac{1025}{14}\right)\sqrt{5}+\dfrac{693}{8}\\
&=&\left(-\dfrac{27143}{280}\right)\sqrt{5}+\dfrac{6709}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{9}{5}\right)\sqrt{45}+\left(-1\right)\sqrt{125}+\dfrac{49}{6}+\left(0\right)\sqrt{25}+\left(-\dfrac{17}{2}\right)\sqrt{125}+\dfrac{26}{9}+\left(\dfrac{49}{8}\right)\sqrt{45}+\left(0\right)\sqrt{45}-\dfrac{9}{2}\right)\times\left(\left(\dfrac{8}{3}\right)\sqrt{45}+\left(-6\right)\sqrt{20}+\left(\dfrac{38}{7}\right)\sqrt{125}-\dfrac{33}{4}+\left(-\dfrac{55}{4}\right)\sqrt{25}+\left(-\dfrac{5}{4}\right)\sqrt{20}-\dfrac{77}{8}+\left(-\dfrac{23}{9}\right)\sqrt{45}+\dfrac{74}{7}-\left(\left(-\dfrac{61}{7}\right)\sqrt{125}\right)-\dfrac{74}{7}-\left(\left(-\dfrac{10}{3}\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\dfrac{27}{5}\right)\sqrt{5}+\left(-5\right)\sqrt{5}+\dfrac{49}{6}+0+\left(-\dfrac{85}{2}\right)\sqrt{5}+\dfrac{26}{9}+\left(\dfrac{147}{8}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{9}{2}\right)\times\left(\left(8\right)\sqrt{5}+\left(-12\right)\sqrt{5}+\left(\dfrac{190}{7}\right)\sqrt{5}-\dfrac{33}{4}-\dfrac{275}{4}+\left(-\dfrac{5}{2}\right)\sqrt{5}-\dfrac{77}{8}+\left(-\dfrac{23}{3}\right)\sqrt{5}+\dfrac{74}{7}-\left(\left(-\dfrac{305}{7}\right)\sqrt{5}\right)-\dfrac{74}{7}-\left(\left(-\dfrac{50}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-\dfrac{949}{40}\right)\sqrt{5}+\dfrac{59}{9}\right)\left(\left(\dfrac{1025}{14}\right)\sqrt{5}-\dfrac{693}{8}\right)\\
&=&\left(-\dfrac{194545}{112}\right)\sqrt{25}+\left(\dfrac{51108391}{20160}\right)\sqrt{5}-\dfrac{4543}{8}\\
&=&-\dfrac{1036327}{112}+\left(\dfrac{51108391}{20160}\right)\sqrt{5}\\
\end{eqnarray*}