L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=2\) et \( Y=\left(\left(-\dfrac{32}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{6}\right)\sqrt{25}\right)+\left(\dfrac{59}{2}\right)\sqrt{20}+\left(-\dfrac{53}{3}\right)\sqrt{20}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(2\right)+\left(\left(\left(-\dfrac{32}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{6}\right)\sqrt{25}\right)+\left(\dfrac{59}{2}\right)\sqrt{20}+\left(-\dfrac{53}{3}\right)\sqrt{20}\right)\\
&=&\left(2\right)+\left(\left(\left(-\dfrac{160}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{115}{6}\right)\sqrt{5}\right)+\dfrac{5}{6}+\left(59\right)\sqrt{5}+\left(-\dfrac{106}{3}\right)\sqrt{5}\right)\\
&=&2+\left(\left(-\dfrac{160}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{115}{6}\right)\sqrt{5}\right)+\dfrac{5}{6}+\left(59\right)\sqrt{5}+\left(-\dfrac{106}{3}\right)\sqrt{5}\\
&=&\dfrac{17}{6}+\left(\dfrac{839}{42}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(2\right)-\left(\left(\left(-\dfrac{32}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{6}\right)\sqrt{25}\right)+\left(\dfrac{59}{2}\right)\sqrt{20}+\left(-\dfrac{53}{3}\right)\sqrt{20}\right)\\
&=&\left(2\right)-\left(\left(\left(-\dfrac{160}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{115}{6}\right)\sqrt{5}\right)+\dfrac{5}{6}+\left(59\right)\sqrt{5}+\left(-\dfrac{106}{3}\right)\sqrt{5}\right)\\
&=&\left(2\right)-\left(\left(\dfrac{839}{42}\right)\sqrt{5}+\dfrac{5}{6}\right)\\
&=&2+\left(-\dfrac{839}{42}\right)\sqrt{5}-\dfrac{5}{6}\\
&=&\dfrac{7}{6}+\left(-\dfrac{839}{42}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(2\right)\times\left(\left(\left(-\dfrac{32}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{125}\right)-\left(\left(-\dfrac{1}{6}\right)\sqrt{25}\right)+\left(\dfrac{59}{2}\right)\sqrt{20}+\left(-\dfrac{53}{3}\right)\sqrt{20}\right)\\
&=&\left(2\right)\times\left(\left(\left(-\dfrac{160}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{115}{6}\right)\sqrt{5}\right)+\dfrac{5}{6}+\left(59\right)\sqrt{5}+\left(-\dfrac{106}{3}\right)\sqrt{5}\right)\\
&=&\left(2\right)\left(\left(\dfrac{839}{42}\right)\sqrt{5}+\dfrac{5}{6}\right)\\
&=&\left(\dfrac{839}{21}\right)\sqrt{5}+\dfrac{5}{3}\\
&=&\left(\dfrac{839}{21}\right)\sqrt{5}+\dfrac{5}{3}\\
\end{eqnarray*}