L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{31}{6}\right)\sqrt{25}+\left(-\dfrac{33}{4}\right)\sqrt{45}+\left(-\dfrac{9}{2}\right)\sqrt{45}+\left(9\right)\sqrt{20}+\dfrac{23}{2}+\left(2\right)\sqrt{20}\) et \( Y=\left(\dfrac{67}{9}\right)\sqrt{20}+\dfrac{8}{3}-\left(\left(\dfrac{30}{7}\right)\sqrt{25}\right)-\left(\left(6\right)\sqrt{25}\right)\) . 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{31}{6}\right)\sqrt{25}+\left(-\dfrac{33}{4}\right)\sqrt{45}+\left(-\dfrac{9}{2}\right)\sqrt{45}+\left(9\right)\sqrt{20}+\dfrac{23}{2}+\left(2\right)\sqrt{20}\right)+\left(\left(\dfrac{67}{9}\right)\sqrt{20}+\dfrac{8}{3}-\left(\left(\dfrac{30}{7}\right)\sqrt{25}\right)-\left(\left(6\right)\sqrt{25}\right)\right)\\
&=&\left(-\dfrac{155}{6}+\left(-\dfrac{99}{4}\right)\sqrt{5}+\left(-\dfrac{27}{2}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\dfrac{23}{2}+\left(4\right)\sqrt{5}\right)+\left(\left(\dfrac{134}{9}\right)\sqrt{5}+\dfrac{8}{3}-\dfrac{150}{7}-30\right)\\
&=&-\dfrac{155}{6}+\left(-\dfrac{99}{4}\right)\sqrt{5}+\left(-\dfrac{27}{2}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\dfrac{23}{2}+\left(4\right)\sqrt{5}+\left(\dfrac{134}{9}\right)\sqrt{5}+\dfrac{8}{3}-\dfrac{150}{7}-30\\
&=&-\dfrac{1325}{21}+\left(-\dfrac{49}{36}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{31}{6}\right)\sqrt{25}+\left(-\dfrac{33}{4}\right)\sqrt{45}+\left(-\dfrac{9}{2}\right)\sqrt{45}+\left(9\right)\sqrt{20}+\dfrac{23}{2}+\left(2\right)\sqrt{20}\right)-\left(\left(\dfrac{67}{9}\right)\sqrt{20}+\dfrac{8}{3}-\left(\left(\dfrac{30}{7}\right)\sqrt{25}\right)-\left(\left(6\right)\sqrt{25}\right)\right)\\
&=&\left(-\dfrac{155}{6}+\left(-\dfrac{99}{4}\right)\sqrt{5}+\left(-\dfrac{27}{2}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\dfrac{23}{2}+\left(4\right)\sqrt{5}\right)-\left(\left(\dfrac{134}{9}\right)\sqrt{5}+\dfrac{8}{3}-\dfrac{150}{7}-30\right)\\
&=&\left(-\dfrac{43}{3}+\left(-\dfrac{65}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{134}{9}\right)\sqrt{5}-\dfrac{1024}{21}\right)\\
&=&-\dfrac{43}{3}+\left(-\dfrac{65}{4}\right)\sqrt{5}+\left(-\dfrac{134}{9}\right)\sqrt{5}+\dfrac{1024}{21}\\
&=&\dfrac{241}{7}+\left(-\dfrac{1121}{36}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{31}{6}\right)\sqrt{25}+\left(-\dfrac{33}{4}\right)\sqrt{45}+\left(-\dfrac{9}{2}\right)\sqrt{45}+\left(9\right)\sqrt{20}+\dfrac{23}{2}+\left(2\right)\sqrt{20}\right)\times\left(\left(\dfrac{67}{9}\right)\sqrt{20}+\dfrac{8}{3}-\left(\left(\dfrac{30}{7}\right)\sqrt{25}\right)-\left(\left(6\right)\sqrt{25}\right)\right)\\
&=&\left(-\dfrac{155}{6}+\left(-\dfrac{99}{4}\right)\sqrt{5}+\left(-\dfrac{27}{2}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\dfrac{23}{2}+\left(4\right)\sqrt{5}\right)\times\left(\left(\dfrac{134}{9}\right)\sqrt{5}+\dfrac{8}{3}-\dfrac{150}{7}-30\right)\\
&=&\left(-\dfrac{43}{3}+\left(-\dfrac{65}{4}\right)\sqrt{5}\right)\left(\left(\dfrac{134}{9}\right)\sqrt{5}-\dfrac{1024}{21}\right)\\
&=&\left(\dfrac{109426}{189}\right)\sqrt{5}+\dfrac{44032}{63}+\left(-\dfrac{4355}{18}\right)\sqrt{25}\\
&=&\left(\dfrac{109426}{189}\right)\sqrt{5}-\dfrac{64361}{126}\\
\end{eqnarray*}