L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{47}{4}\) et \( Y=\left(\dfrac{7}{9}-\left(\left(\dfrac{26}{7}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{20}+\left(-9\right)\sqrt{125}-\dfrac{41}{2}+\left(2\right)\sqrt{125}\right)-\left(\left(-7\right)\sqrt{20}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{45}\right)-\left(\left(\left(-9\right)\sqrt{45}\right)-\left(\left(\dfrac{13}{9}\right)\sqrt{20}\right)+\dfrac{67}{7}-\left(\left(8\right)\sqrt{25}\right)-9\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{47}{4}\right)+\left(\left(\dfrac{7}{9}-\left(\left(\dfrac{26}{7}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{20}+\left(-9\right)\sqrt{125}-\dfrac{41}{2}+\left(2\right)\sqrt{125}\right)-\left(\left(-7\right)\sqrt{20}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{45}\right)-\left(\left(\left(-9\right)\sqrt{45}\right)-\left(\left(\dfrac{13}{9}\right)\sqrt{20}\right)+\dfrac{67}{7}-\left(\left(8\right)\sqrt{25}\right)-9\right)\right)\\
&=&\left(\dfrac{47}{4}\right)+\left(\left(\dfrac{7}{9}-\left(\left(\dfrac{52}{7}\right)\sqrt{5}\right)\right)-\left(\left(19\right)\sqrt{5}+\left(-45\right)\sqrt{5}-\dfrac{41}{2}+\left(10\right)\sqrt{5}\right)-\left(\left(-14\right)\sqrt{5}\right)-\left(\left(-\dfrac{189}{4}\right)\sqrt{5}\right)-\left(\left(\left(-27\right)\sqrt{5}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{5}\right)+\dfrac{67}{7}-40-9\right)\right)\\
&=&\dfrac{47}{4}+\left(\dfrac{7}{9}-\left(\left(\dfrac{52}{7}\right)\sqrt{5}\right)\right)-\left(\left(19\right)\sqrt{5}+\left(-45\right)\sqrt{5}-\dfrac{41}{2}+\left(10\right)\sqrt{5}\right)-\left(\left(-14\right)\sqrt{5}\right)-\left(\left(-\dfrac{189}{4}\right)\sqrt{5}\right)-\left(\left(\left(-27\right)\sqrt{5}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{5}\right)+\dfrac{67}{7}-40-9\right)\\
&=&\dfrac{18259}{252}+\left(\dfrac{25127}{252}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{47}{4}\right)-\left(\left(\dfrac{7}{9}-\left(\left(\dfrac{26}{7}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{20}+\left(-9\right)\sqrt{125}-\dfrac{41}{2}+\left(2\right)\sqrt{125}\right)-\left(\left(-7\right)\sqrt{20}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{45}\right)-\left(\left(\left(-9\right)\sqrt{45}\right)-\left(\left(\dfrac{13}{9}\right)\sqrt{20}\right)+\dfrac{67}{7}-\left(\left(8\right)\sqrt{25}\right)-9\right)\right)\\
&=&\left(\dfrac{47}{4}\right)-\left(\left(\dfrac{7}{9}-\left(\left(\dfrac{52}{7}\right)\sqrt{5}\right)\right)-\left(\left(19\right)\sqrt{5}+\left(-45\right)\sqrt{5}-\dfrac{41}{2}+\left(10\right)\sqrt{5}\right)-\left(\left(-14\right)\sqrt{5}\right)-\left(\left(-\dfrac{189}{4}\right)\sqrt{5}\right)-\left(\left(\left(-27\right)\sqrt{5}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{5}\right)+\dfrac{67}{7}-40-9\right)\right)\\
&=&\left(\dfrac{47}{4}\right)-\left(\dfrac{7649}{126}+\left(\dfrac{25127}{252}\right)\sqrt{5}\right)\\
&=&\dfrac{47}{4}+-\dfrac{7649}{126}+\left(-\dfrac{25127}{252}\right)\sqrt{5}\\
&=&-\dfrac{12337}{252}+\left(-\dfrac{25127}{252}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{47}{4}\right)\times\left(\left(\dfrac{7}{9}-\left(\left(\dfrac{26}{7}\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{20}+\left(-9\right)\sqrt{125}-\dfrac{41}{2}+\left(2\right)\sqrt{125}\right)-\left(\left(-7\right)\sqrt{20}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{45}\right)-\left(\left(\left(-9\right)\sqrt{45}\right)-\left(\left(\dfrac{13}{9}\right)\sqrt{20}\right)+\dfrac{67}{7}-\left(\left(8\right)\sqrt{25}\right)-9\right)\right)\\
&=&\left(\dfrac{47}{4}\right)\times\left(\left(\dfrac{7}{9}-\left(\left(\dfrac{52}{7}\right)\sqrt{5}\right)\right)-\left(\left(19\right)\sqrt{5}+\left(-45\right)\sqrt{5}-\dfrac{41}{2}+\left(10\right)\sqrt{5}\right)-\left(\left(-14\right)\sqrt{5}\right)-\left(\left(-\dfrac{189}{4}\right)\sqrt{5}\right)-\left(\left(\left(-27\right)\sqrt{5}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{5}\right)+\dfrac{67}{7}-40-9\right)\right)\\
&=&\left(\dfrac{47}{4}\right)\left(\dfrac{7649}{126}+\left(\dfrac{25127}{252}\right)\sqrt{5}\right)\\
&=&\dfrac{359503}{504}+\left(\dfrac{1180969}{1008}\right)\sqrt{5}\\
&=&\dfrac{359503}{504}+\left(\dfrac{1180969}{1008}\right)\sqrt{5}\\
\end{eqnarray*}