L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{73}{4}+\left(-\dfrac{14}{5}\right)\sqrt{45}+\left(-\dfrac{17}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{63}{5}\right)\sqrt{20}\right)\) et \( Y=\left(\dfrac{40}{9}\right)\sqrt{20}+\left(0\right)\sqrt{45}+\left(\left(-\dfrac{28}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{14}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)-\left(\left(9\right)\sqrt{20}\right)+\left(-7\right)\sqrt{20}+\left(-\dfrac{71}{7}\right)\sqrt{25}+\left(-\dfrac{71}{7}\right)\sqrt{25}+\left(5\right)\sqrt{45}+\dfrac{20}{7}-\dfrac{25}{4}\) . 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{73}{4}+\left(-\dfrac{14}{5}\right)\sqrt{45}+\left(-\dfrac{17}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{63}{5}\right)\sqrt{20}\right)\right)+\left(\left(\dfrac{40}{9}\right)\sqrt{20}+\left(0\right)\sqrt{45}+\left(\left(-\dfrac{28}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{14}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)-\left(\left(9\right)\sqrt{20}\right)+\left(-7\right)\sqrt{20}+\left(-\dfrac{71}{7}\right)\sqrt{25}+\left(-\dfrac{71}{7}\right)\sqrt{25}+\left(5\right)\sqrt{45}+\dfrac{20}{7}-\dfrac{25}{4}\right)\\
&=&\left(\left(\dfrac{73}{4}+\left(-\dfrac{42}{5}\right)\sqrt{5}+\left(-\dfrac{17}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{126}{5}\right)\sqrt{5}\right)\right)+\left(\left(\dfrac{80}{9}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{140}{9}-14-\left(\left(24\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)+\left(-14\right)\sqrt{5}-\dfrac{355}{7}-\dfrac{355}{7}+\left(15\right)\sqrt{5}+\dfrac{20}{7}-\dfrac{25}{4}\right)\\
&=&\left(\dfrac{73}{4}+\left(-\dfrac{42}{5}\right)\sqrt{5}+\left(-\dfrac{17}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{126}{5}\right)\sqrt{5}\right)+\left(\dfrac{80}{9}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{140}{9}-14-\left(\left(24\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)+\left(-14\right)\sqrt{5}-\dfrac{355}{7}-\dfrac{355}{7}+\left(15\right)\sqrt{5}+\dfrac{20}{7}-\dfrac{25}{4}\\
&=&-\dfrac{7316}{63}+\left(-\dfrac{6679}{90}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{73}{4}+\left(-\dfrac{14}{5}\right)\sqrt{45}+\left(-\dfrac{17}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{63}{5}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{40}{9}\right)\sqrt{20}+\left(0\right)\sqrt{45}+\left(\left(-\dfrac{28}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{14}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)-\left(\left(9\right)\sqrt{20}\right)+\left(-7\right)\sqrt{20}+\left(-\dfrac{71}{7}\right)\sqrt{25}+\left(-\dfrac{71}{7}\right)\sqrt{25}+\left(5\right)\sqrt{45}+\dfrac{20}{7}-\dfrac{25}{4}\right)\\
&=&\left(\left(\dfrac{73}{4}+\left(-\dfrac{42}{5}\right)\sqrt{5}+\left(-\dfrac{17}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{126}{5}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{80}{9}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{140}{9}-14-\left(\left(24\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)+\left(-14\right)\sqrt{5}-\dfrac{355}{7}-\dfrac{355}{7}+\left(15\right)\sqrt{5}+\dfrac{20}{7}-\dfrac{25}{4}\right)\\
&=&\left(\dfrac{73}{4}+\left(-\dfrac{421}{10}\right)\sqrt{5}\right)-\left(\left(-\dfrac{289}{9}\right)\sqrt{5}-\dfrac{33863}{252}\right)\\
&=&\dfrac{73}{4}+\left(-\dfrac{421}{10}\right)\sqrt{5}+\left(\dfrac{289}{9}\right)\sqrt{5}+\dfrac{33863}{252}\\
&=&\dfrac{19231}{126}+\left(-\dfrac{899}{90}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{73}{4}+\left(-\dfrac{14}{5}\right)\sqrt{45}+\left(-\dfrac{17}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{63}{5}\right)\sqrt{20}\right)\right)\times\left(\left(\dfrac{40}{9}\right)\sqrt{20}+\left(0\right)\sqrt{45}+\left(\left(-\dfrac{28}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{14}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)-\left(\left(9\right)\sqrt{20}\right)+\left(-7\right)\sqrt{20}+\left(-\dfrac{71}{7}\right)\sqrt{25}+\left(-\dfrac{71}{7}\right)\sqrt{25}+\left(5\right)\sqrt{45}+\dfrac{20}{7}-\dfrac{25}{4}\right)\\
&=&\left(\left(\dfrac{73}{4}+\left(-\dfrac{42}{5}\right)\sqrt{5}+\left(-\dfrac{17}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{126}{5}\right)\sqrt{5}\right)\right)\times\left(\left(\dfrac{80}{9}\right)\sqrt{5}+\left(0\right)\sqrt{5}-\dfrac{140}{9}-14-\left(\left(24\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)+\left(-14\right)\sqrt{5}-\dfrac{355}{7}-\dfrac{355}{7}+\left(15\right)\sqrt{5}+\dfrac{20}{7}-\dfrac{25}{4}\right)\\
&=&\left(\dfrac{73}{4}+\left(-\dfrac{421}{10}\right)\sqrt{5}\right)\left(\left(-\dfrac{289}{9}\right)\sqrt{5}-\dfrac{33863}{252}\right)\\
&=&\left(\dfrac{12779533}{2520}\right)\sqrt{5}-\dfrac{2471999}{1008}+\left(\dfrac{121669}{90}\right)\sqrt{25}\\
&=&\left(\dfrac{12779533}{2520}\right)\sqrt{5}+\dfrac{482385}{112}\\
\end{eqnarray*}