L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(6\right)\sqrt{25}+\left(-\dfrac{17}{2}\right)\sqrt{125}+\dfrac{34}{5}+\left(7\right)\sqrt{45}+\left(\left(-\dfrac{27}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{73}{7}\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{25}\right)-\left(\left(\dfrac{81}{5}\right)\sqrt{45}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{25}\right)\) et \( Y=\left(-\dfrac{47}{6}\right)\sqrt{125}+\left(-\dfrac{1}{3}\right)\sqrt{45}+\left(\left(-\dfrac{79}{9}\right)\sqrt{25}\right)-\left(\left(0\right)\sqrt{20}\right)-\left(\left(9\right)\sqrt{20}\right)+\left(\dfrac{48}{7}\right)\sqrt{20}+\dfrac{67}{4}+\left(\dfrac{2}{3}\right)\sqrt{20}+\left(-\dfrac{15}{2}\right)\sqrt{20}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(6\right)\sqrt{25}+\left(-\dfrac{17}{2}\right)\sqrt{125}+\dfrac{34}{5}+\left(7\right)\sqrt{45}+\left(\left(-\dfrac{27}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{73}{7}\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{25}\right)-\left(\left(\dfrac{81}{5}\right)\sqrt{45}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{25}\right)\right)+\left(\left(-\dfrac{47}{6}\right)\sqrt{125}+\left(-\dfrac{1}{3}\right)\sqrt{45}+\left(\left(-\dfrac{79}{9}\right)\sqrt{25}\right)-\left(\left(0\right)\sqrt{20}\right)-\left(\left(9\right)\sqrt{20}\right)+\left(\dfrac{48}{7}\right)\sqrt{20}+\dfrac{67}{4}+\left(\dfrac{2}{3}\right)\sqrt{20}+\left(-\dfrac{15}{2}\right)\sqrt{20}\right)\\
&=&\left(30+\left(-\dfrac{85}{2}\right)\sqrt{5}+\dfrac{34}{5}+\left(21\right)\sqrt{5}+\left(\left(-\dfrac{135}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{146}{7}\right)\sqrt{5}\right)-30-\left(\left(\dfrac{243}{5}\right)\sqrt{5}\right)+48\right)+\left(\left(-\dfrac{235}{6}\right)\sqrt{5}+\left(-1\right)\sqrt{5}-\dfrac{395}{9}-\left(\left(0\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)+\left(\dfrac{96}{7}\right)\sqrt{5}+\dfrac{67}{4}+\left(\dfrac{4}{3}\right)\sqrt{5}+\left(-15\right)\sqrt{5}\right)\\
&=&30+\left(-\dfrac{85}{2}\right)\sqrt{5}+\dfrac{34}{5}+\left(21\right)\sqrt{5}+\left(\left(-\dfrac{135}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{146}{7}\right)\sqrt{5}\right)-30-\left(\left(\dfrac{243}{5}\right)\sqrt{5}\right)+48+\left(-\dfrac{235}{6}\right)\sqrt{5}+\left(-1\right)\sqrt{5}-\dfrac{395}{9}-\left(\left(0\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)+\left(\dfrac{96}{7}\right)\sqrt{5}+\dfrac{67}{4}+\left(\dfrac{4}{3}\right)\sqrt{5}+\left(-15\right)\sqrt{5}\\
&=&\dfrac{4979}{180}+\left(-\dfrac{45481}{210}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(6\right)\sqrt{25}+\left(-\dfrac{17}{2}\right)\sqrt{125}+\dfrac{34}{5}+\left(7\right)\sqrt{45}+\left(\left(-\dfrac{27}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{73}{7}\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{25}\right)-\left(\left(\dfrac{81}{5}\right)\sqrt{45}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{25}\right)\right)-\left(\left(-\dfrac{47}{6}\right)\sqrt{125}+\left(-\dfrac{1}{3}\right)\sqrt{45}+\left(\left(-\dfrac{79}{9}\right)\sqrt{25}\right)-\left(\left(0\right)\sqrt{20}\right)-\left(\left(9\right)\sqrt{20}\right)+\left(\dfrac{48}{7}\right)\sqrt{20}+\dfrac{67}{4}+\left(\dfrac{2}{3}\right)\sqrt{20}+\left(-\dfrac{15}{2}\right)\sqrt{20}\right)\\
&=&\left(30+\left(-\dfrac{85}{2}\right)\sqrt{5}+\dfrac{34}{5}+\left(21\right)\sqrt{5}+\left(\left(-\dfrac{135}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{146}{7}\right)\sqrt{5}\right)-30-\left(\left(\dfrac{243}{5}\right)\sqrt{5}\right)+48\right)-\left(\left(-\dfrac{235}{6}\right)\sqrt{5}+\left(-1\right)\sqrt{5}-\dfrac{395}{9}-\left(\left(0\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)+\left(\dfrac{96}{7}\right)\sqrt{5}+\dfrac{67}{4}+\left(\dfrac{4}{3}\right)\sqrt{5}+\left(-15\right)\sqrt{5}\right)\\
&=&\left(\dfrac{274}{5}+\left(-\dfrac{5546}{35}\right)\sqrt{5}\right)-\left(\left(-\dfrac{2441}{42}\right)\sqrt{5}-\dfrac{977}{36}\right)\\
&=&\dfrac{274}{5}+\left(-\dfrac{5546}{35}\right)\sqrt{5}+\left(\dfrac{2441}{42}\right)\sqrt{5}+\dfrac{977}{36}\\
&=&\dfrac{14749}{180}+\left(-\dfrac{21071}{210}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(6\right)\sqrt{25}+\left(-\dfrac{17}{2}\right)\sqrt{125}+\dfrac{34}{5}+\left(7\right)\sqrt{45}+\left(\left(-\dfrac{27}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{73}{7}\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{25}\right)-\left(\left(\dfrac{81}{5}\right)\sqrt{45}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{25}\right)\right)\times\left(\left(-\dfrac{47}{6}\right)\sqrt{125}+\left(-\dfrac{1}{3}\right)\sqrt{45}+\left(\left(-\dfrac{79}{9}\right)\sqrt{25}\right)-\left(\left(0\right)\sqrt{20}\right)-\left(\left(9\right)\sqrt{20}\right)+\left(\dfrac{48}{7}\right)\sqrt{20}+\dfrac{67}{4}+\left(\dfrac{2}{3}\right)\sqrt{20}+\left(-\dfrac{15}{2}\right)\sqrt{20}\right)\\
&=&\left(30+\left(-\dfrac{85}{2}\right)\sqrt{5}+\dfrac{34}{5}+\left(21\right)\sqrt{5}+\left(\left(-\dfrac{135}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{146}{7}\right)\sqrt{5}\right)-30-\left(\left(\dfrac{243}{5}\right)\sqrt{5}\right)+48\right)\times\left(\left(-\dfrac{235}{6}\right)\sqrt{5}+\left(-1\right)\sqrt{5}-\dfrac{395}{9}-\left(\left(0\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)+\left(\dfrac{96}{7}\right)\sqrt{5}+\dfrac{67}{4}+\left(\dfrac{4}{3}\right)\sqrt{5}+\left(-15\right)\sqrt{5}\right)\\
&=&\left(\dfrac{274}{5}+\left(-\dfrac{5546}{35}\right)\sqrt{5}\right)\left(\left(-\dfrac{2441}{42}\right)\sqrt{5}-\dfrac{977}{36}\right)\\
&=&\left(\dfrac{702719}{630}\right)\sqrt{5}-\dfrac{133849}{90}+\left(\dfrac{6768893}{735}\right)\sqrt{25}\\
&=&\left(\dfrac{702719}{630}\right)\sqrt{5}+\dfrac{196508189}{4410}\\
\end{eqnarray*}