L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{52}{3}\right)\sqrt{20}+\left(-\dfrac{17}{2}\right)\sqrt{45}+\left(-\dfrac{43}{7}\right)\sqrt{20}+\left(8\right)\sqrt{45}\) et \( Y=\left(\left(-\dfrac{23}{4}\right)\sqrt{45}\right)+3-\left(\left(\dfrac{76}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{7}\right)\sqrt{25}\right)-6-\left(\left(-\dfrac{52}{3}\right)\sqrt{20}\right)+\left(-\dfrac{21}{4}\right)\sqrt{20}+\left(-\dfrac{21}{8}\right)\sqrt{125}+\left(\dfrac{43}{6}\right)\sqrt{20}+\left(\dfrac{32}{7}\right)\sqrt{45}+\left(-\dfrac{31}{2}\right)\sqrt{125}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{52}{3}\right)\sqrt{20}+\left(-\dfrac{17}{2}\right)\sqrt{45}+\left(-\dfrac{43}{7}\right)\sqrt{20}+\left(8\right)\sqrt{45}\right)+\left(\left(\left(-\dfrac{23}{4}\right)\sqrt{45}\right)+3-\left(\left(\dfrac{76}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{7}\right)\sqrt{25}\right)-6-\left(\left(-\dfrac{52}{3}\right)\sqrt{20}\right)+\left(-\dfrac{21}{4}\right)\sqrt{20}+\left(-\dfrac{21}{8}\right)\sqrt{125}+\left(\dfrac{43}{6}\right)\sqrt{20}+\left(\dfrac{32}{7}\right)\sqrt{45}+\left(-\dfrac{31}{2}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{104}{3}\right)\sqrt{5}+\left(-\dfrac{51}{2}\right)\sqrt{5}+\left(-\dfrac{86}{7}\right)\sqrt{5}+\left(24\right)\sqrt{5}\right)+\left(\left(\left(-\dfrac{69}{4}\right)\sqrt{5}\right)+3-\left(\left(\dfrac{76}{3}\right)\sqrt{5}\right)+\dfrac{15}{7}-6-\left(\left(-\dfrac{104}{3}\right)\sqrt{5}\right)+\left(-\dfrac{21}{2}\right)\sqrt{5}+\left(-\dfrac{105}{8}\right)\sqrt{5}+\left(\dfrac{43}{3}\right)\sqrt{5}+\left(\dfrac{96}{7}\right)\sqrt{5}+\left(-\dfrac{155}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{104}{3}\right)\sqrt{5}+\left(-\dfrac{51}{2}\right)\sqrt{5}+\left(-\dfrac{86}{7}\right)\sqrt{5}+\left(24\right)\sqrt{5}+\left(\left(-\dfrac{69}{4}\right)\sqrt{5}\right)+3-\left(\left(\dfrac{76}{3}\right)\sqrt{5}\right)+\dfrac{15}{7}-6-\left(\left(-\dfrac{104}{3}\right)\sqrt{5}\right)+\left(-\dfrac{21}{2}\right)\sqrt{5}+\left(-\dfrac{105}{8}\right)\sqrt{5}+\left(\dfrac{43}{3}\right)\sqrt{5}+\left(\dfrac{96}{7}\right)\sqrt{5}+\left(-\dfrac{155}{2}\right)\sqrt{5}\\
&=&\left(-\dfrac{10099}{168}\right)\sqrt{5}-\dfrac{6}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{52}{3}\right)\sqrt{20}+\left(-\dfrac{17}{2}\right)\sqrt{45}+\left(-\dfrac{43}{7}\right)\sqrt{20}+\left(8\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{23}{4}\right)\sqrt{45}\right)+3-\left(\left(\dfrac{76}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{7}\right)\sqrt{25}\right)-6-\left(\left(-\dfrac{52}{3}\right)\sqrt{20}\right)+\left(-\dfrac{21}{4}\right)\sqrt{20}+\left(-\dfrac{21}{8}\right)\sqrt{125}+\left(\dfrac{43}{6}\right)\sqrt{20}+\left(\dfrac{32}{7}\right)\sqrt{45}+\left(-\dfrac{31}{2}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{104}{3}\right)\sqrt{5}+\left(-\dfrac{51}{2}\right)\sqrt{5}+\left(-\dfrac{86}{7}\right)\sqrt{5}+\left(24\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{69}{4}\right)\sqrt{5}\right)+3-\left(\left(\dfrac{76}{3}\right)\sqrt{5}\right)+\dfrac{15}{7}-6-\left(\left(-\dfrac{104}{3}\right)\sqrt{5}\right)+\left(-\dfrac{21}{2}\right)\sqrt{5}+\left(-\dfrac{105}{8}\right)\sqrt{5}+\left(\dfrac{43}{3}\right)\sqrt{5}+\left(\dfrac{96}{7}\right)\sqrt{5}+\left(-\dfrac{155}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{877}{42}\right)\sqrt{5}\right)-\left(\left(-\dfrac{13607}{168}\right)\sqrt{5}-\dfrac{6}{7}\right)\\
&=&\left(\dfrac{877}{42}\right)\sqrt{5}+\left(\dfrac{13607}{168}\right)\sqrt{5}+\dfrac{6}{7}\\
&=&\left(\dfrac{815}{8}\right)\sqrt{5}+\dfrac{6}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{52}{3}\right)\sqrt{20}+\left(-\dfrac{17}{2}\right)\sqrt{45}+\left(-\dfrac{43}{7}\right)\sqrt{20}+\left(8\right)\sqrt{45}\right)\times\left(\left(\left(-\dfrac{23}{4}\right)\sqrt{45}\right)+3-\left(\left(\dfrac{76}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{7}\right)\sqrt{25}\right)-6-\left(\left(-\dfrac{52}{3}\right)\sqrt{20}\right)+\left(-\dfrac{21}{4}\right)\sqrt{20}+\left(-\dfrac{21}{8}\right)\sqrt{125}+\left(\dfrac{43}{6}\right)\sqrt{20}+\left(\dfrac{32}{7}\right)\sqrt{45}+\left(-\dfrac{31}{2}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{104}{3}\right)\sqrt{5}+\left(-\dfrac{51}{2}\right)\sqrt{5}+\left(-\dfrac{86}{7}\right)\sqrt{5}+\left(24\right)\sqrt{5}\right)\times\left(\left(\left(-\dfrac{69}{4}\right)\sqrt{5}\right)+3-\left(\left(\dfrac{76}{3}\right)\sqrt{5}\right)+\dfrac{15}{7}-6-\left(\left(-\dfrac{104}{3}\right)\sqrt{5}\right)+\left(-\dfrac{21}{2}\right)\sqrt{5}+\left(-\dfrac{105}{8}\right)\sqrt{5}+\left(\dfrac{43}{3}\right)\sqrt{5}+\left(\dfrac{96}{7}\right)\sqrt{5}+\left(-\dfrac{155}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{877}{42}\right)\sqrt{5}\right)\left(\left(-\dfrac{13607}{168}\right)\sqrt{5}-\dfrac{6}{7}\right)\\
&=&\left(-\dfrac{11933339}{7056}\right)\sqrt{25}+\left(-\dfrac{877}{49}\right)\sqrt{5}\\
&=&-\dfrac{59666695}{7056}+\left(-\dfrac{877}{49}\right)\sqrt{5}\\
\end{eqnarray*}