L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{23}{7}+\left(-1\right)\sqrt{50}\right)-\left(\left(-6\right)\sqrt{8}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{8}\right)\) et \( Y=\left(\left(\dfrac{47}{7}\right)\sqrt{50}+\left(\dfrac{61}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{37}{4}\right)\sqrt{4}\right)\) . 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{23}{7}+\left(-1\right)\sqrt{50}\right)-\left(\left(-6\right)\sqrt{8}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{8}\right)\right)+\left(\left(\left(\dfrac{47}{7}\right)\sqrt{50}+\left(\dfrac{61}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{37}{4}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(-\dfrac{23}{7}+\left(-5\right)\sqrt{2}\right)-\left(\left(-12\right)\sqrt{2}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{2}\right)\right)+\left(\left(\left(\dfrac{235}{7}\right)\sqrt{2}+\left(\dfrac{305}{8}\right)\sqrt{2}\right)+\dfrac{37}{2}\right)\\
&=&\left(-\dfrac{23}{7}+\left(-5\right)\sqrt{2}\right)-\left(\left(-12\right)\sqrt{2}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{2}\right)+\left(\left(\dfrac{235}{7}\right)\sqrt{2}+\left(\dfrac{305}{8}\right)\sqrt{2}\right)+\dfrac{37}{2}\\
&=&\dfrac{213}{14}+\left(\dfrac{15685}{168}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{23}{7}+\left(-1\right)\sqrt{50}\right)-\left(\left(-6\right)\sqrt{8}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{8}\right)\right)-\left(\left(\left(\dfrac{47}{7}\right)\sqrt{50}+\left(\dfrac{61}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{37}{4}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(-\dfrac{23}{7}+\left(-5\right)\sqrt{2}\right)-\left(\left(-12\right)\sqrt{2}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{2}\right)\right)-\left(\left(\left(\dfrac{235}{7}\right)\sqrt{2}+\left(\dfrac{305}{8}\right)\sqrt{2}\right)+\dfrac{37}{2}\right)\\
&=&\left(-\dfrac{23}{7}+\left(\dfrac{65}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{4015}{56}\right)\sqrt{2}+\dfrac{37}{2}\right)\\
&=&-\dfrac{23}{7}+\left(\dfrac{65}{3}\right)\sqrt{2}+\left(-\dfrac{4015}{56}\right)\sqrt{2}-\dfrac{37}{2}\\
&=&-\dfrac{305}{14}+\left(-\dfrac{8405}{168}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{23}{7}+\left(-1\right)\sqrt{50}\right)-\left(\left(-6\right)\sqrt{8}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{8}\right)\right)\times\left(\left(\left(\dfrac{47}{7}\right)\sqrt{50}+\left(\dfrac{61}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{37}{4}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(-\dfrac{23}{7}+\left(-5\right)\sqrt{2}\right)-\left(\left(-12\right)\sqrt{2}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{2}\right)\right)\times\left(\left(\left(\dfrac{235}{7}\right)\sqrt{2}+\left(\dfrac{305}{8}\right)\sqrt{2}\right)+\dfrac{37}{2}\right)\\
&=&\left(-\dfrac{23}{7}+\left(\dfrac{65}{3}\right)\sqrt{2}\right)\left(\left(\dfrac{4015}{56}\right)\sqrt{2}+\dfrac{37}{2}\right)\\
&=&\left(\dfrac{194345}{1176}\right)\sqrt{2}-\dfrac{851}{14}+\left(\dfrac{260975}{168}\right)\sqrt{4}\\
&=&\left(\dfrac{194345}{1176}\right)\sqrt{2}+\dfrac{255869}{84}\\
\end{eqnarray*}