L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=1+\left(-\dfrac{58}{7}\right)\sqrt{45}+\left(\dfrac{3}{2}\right)\sqrt{125}+\left(3\right)\sqrt{45}-\dfrac{19}{2}+\dfrac{52}{9}\) et \( Y=\left(-\dfrac{76}{7}\right)\sqrt{20}+\left(\left(-\dfrac{79}{2}\right)\sqrt{20}\right)-\left(\left(-4\right)\sqrt{45}\right)+\dfrac{3}{5}+\left(-1\right)\sqrt{125}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(1+\left(-\dfrac{58}{7}\right)\sqrt{45}+\left(\dfrac{3}{2}\right)\sqrt{125}+\left(3\right)\sqrt{45}-\dfrac{19}{2}+\dfrac{52}{9}\right)+\left(\left(-\dfrac{76}{7}\right)\sqrt{20}+\left(\left(-\dfrac{79}{2}\right)\sqrt{20}\right)-\left(\left(-4\right)\sqrt{45}\right)+\dfrac{3}{5}+\left(-1\right)\sqrt{125}\right)\\
&=&\left(1+\left(-\dfrac{174}{7}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}+\left(9\right)\sqrt{5}-\dfrac{19}{2}+\dfrac{52}{9}\right)+\left(\left(-\dfrac{152}{7}\right)\sqrt{5}+\left(\left(-79\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)+\dfrac{3}{5}+\left(-5\right)\sqrt{5}\right)\\
&=&1+\left(-\dfrac{174}{7}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}+\left(9\right)\sqrt{5}-\dfrac{19}{2}+\dfrac{52}{9}+\left(-\dfrac{152}{7}\right)\sqrt{5}+\left(\left(-79\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)+\dfrac{3}{5}+\left(-5\right)\sqrt{5}\\
&=&-\dfrac{191}{90}+\left(-\dfrac{1429}{14}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(1+\left(-\dfrac{58}{7}\right)\sqrt{45}+\left(\dfrac{3}{2}\right)\sqrt{125}+\left(3\right)\sqrt{45}-\dfrac{19}{2}+\dfrac{52}{9}\right)-\left(\left(-\dfrac{76}{7}\right)\sqrt{20}+\left(\left(-\dfrac{79}{2}\right)\sqrt{20}\right)-\left(\left(-4\right)\sqrt{45}\right)+\dfrac{3}{5}+\left(-1\right)\sqrt{125}\right)\\
&=&\left(1+\left(-\dfrac{174}{7}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}+\left(9\right)\sqrt{5}-\dfrac{19}{2}+\dfrac{52}{9}\right)-\left(\left(-\dfrac{152}{7}\right)\sqrt{5}+\left(\left(-79\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)+\dfrac{3}{5}+\left(-5\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{49}{18}+\left(-\dfrac{117}{14}\right)\sqrt{5}\right)-\left(\left(-\dfrac{656}{7}\right)\sqrt{5}+\dfrac{3}{5}\right)\\
&=&-\dfrac{49}{18}+\left(-\dfrac{117}{14}\right)\sqrt{5}+\left(\dfrac{656}{7}\right)\sqrt{5}-\dfrac{3}{5}\\
&=&-\dfrac{299}{90}+\left(\dfrac{1195}{14}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(1+\left(-\dfrac{58}{7}\right)\sqrt{45}+\left(\dfrac{3}{2}\right)\sqrt{125}+\left(3\right)\sqrt{45}-\dfrac{19}{2}+\dfrac{52}{9}\right)\times\left(\left(-\dfrac{76}{7}\right)\sqrt{20}+\left(\left(-\dfrac{79}{2}\right)\sqrt{20}\right)-\left(\left(-4\right)\sqrt{45}\right)+\dfrac{3}{5}+\left(-1\right)\sqrt{125}\right)\\
&=&\left(1+\left(-\dfrac{174}{7}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}+\left(9\right)\sqrt{5}-\dfrac{19}{2}+\dfrac{52}{9}\right)\times\left(\left(-\dfrac{152}{7}\right)\sqrt{5}+\left(\left(-79\right)\sqrt{5}\right)-\left(\left(-12\right)\sqrt{5}\right)+\dfrac{3}{5}+\left(-5\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{49}{18}+\left(-\dfrac{117}{14}\right)\sqrt{5}\right)\left(\left(-\dfrac{656}{7}\right)\sqrt{5}+\dfrac{3}{5}\right)\\
&=&\left(\dfrac{157561}{630}\right)\sqrt{5}-\dfrac{49}{30}+\left(\dfrac{38376}{49}\right)\sqrt{25}\\
&=&\left(\dfrac{157561}{630}\right)\sqrt{5}+\dfrac{5753999}{1470}\\
\end{eqnarray*}