L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{39}{4}\right)\sqrt{45}\) et \( Y=\left(\left(-2\right)\sqrt{125}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{45}+\left(\dfrac{59}{2}\right)\sqrt{25}+\left(2\right)\sqrt{25}+\left(3\right)\sqrt{45}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{125}\right)-\dfrac{8}{3}\) . 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{39}{4}\right)\sqrt{45}\right)+\left(\left(\left(-2\right)\sqrt{125}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{45}+\left(\dfrac{59}{2}\right)\sqrt{25}+\left(2\right)\sqrt{25}+\left(3\right)\sqrt{45}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{125}\right)-\dfrac{8}{3}\right)\\
&=&\left(\left(\dfrac{117}{4}\right)\sqrt{5}\right)+\left(\left(\left(-10\right)\sqrt{5}\right)-\left(\left(\dfrac{59}{2}\right)\sqrt{5}+\dfrac{295}{2}+10+\left(9\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{4}\right)\sqrt{5}\right)-\dfrac{8}{3}\right)\\
&=&\left(\dfrac{117}{4}\right)\sqrt{5}+\left(\left(-10\right)\sqrt{5}\right)-\left(\left(\dfrac{59}{2}\right)\sqrt{5}+\dfrac{295}{2}+10+\left(9\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{4}\right)\sqrt{5}\right)-\dfrac{8}{3}\\
&=&\left(-58\right)\sqrt{5}-\dfrac{961}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{39}{4}\right)\sqrt{45}\right)-\left(\left(\left(-2\right)\sqrt{125}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{45}+\left(\dfrac{59}{2}\right)\sqrt{25}+\left(2\right)\sqrt{25}+\left(3\right)\sqrt{45}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{125}\right)-\dfrac{8}{3}\right)\\
&=&\left(\left(\dfrac{117}{4}\right)\sqrt{5}\right)-\left(\left(\left(-10\right)\sqrt{5}\right)-\left(\left(\dfrac{59}{2}\right)\sqrt{5}+\dfrac{295}{2}+10+\left(9\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{4}\right)\sqrt{5}\right)-\dfrac{8}{3}\right)\\
&=&\left(\left(\dfrac{117}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{349}{4}\right)\sqrt{5}-\dfrac{961}{6}\right)\\
&=&\left(\dfrac{117}{4}\right)\sqrt{5}+\left(\dfrac{349}{4}\right)\sqrt{5}+\dfrac{961}{6}\\
&=&\left(\dfrac{233}{2}\right)\sqrt{5}+\dfrac{961}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{39}{4}\right)\sqrt{45}\right)\times\left(\left(\left(-2\right)\sqrt{125}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{45}+\left(\dfrac{59}{2}\right)\sqrt{25}+\left(2\right)\sqrt{25}+\left(3\right)\sqrt{45}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{125}\right)-\dfrac{8}{3}\right)\\
&=&\left(\left(\dfrac{117}{4}\right)\sqrt{5}\right)\times\left(\left(\left(-10\right)\sqrt{5}\right)-\left(\left(\dfrac{59}{2}\right)\sqrt{5}+\dfrac{295}{2}+10+\left(9\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{4}\right)\sqrt{5}\right)-\dfrac{8}{3}\right)\\
&=&\left(\left(\dfrac{117}{4}\right)\sqrt{5}\right)\left(\left(-\dfrac{349}{4}\right)\sqrt{5}-\dfrac{961}{6}\right)\\
&=&\left(-\dfrac{40833}{16}\right)\sqrt{25}+\left(-\dfrac{37479}{8}\right)\sqrt{5}\\
&=&-\dfrac{204165}{16}+\left(-\dfrac{37479}{8}\right)\sqrt{5}\\
\end{eqnarray*}