L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{69}{7}\right)\sqrt{20}\) et \( Y=\left(\left(6\right)\sqrt{125}+\left(-\dfrac{18}{7}\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{13}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{59}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{17}{8}\right)\sqrt{45}\right)\right)\) . 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{69}{7}\right)\sqrt{20}\right)+\left(\left(\left(6\right)\sqrt{125}+\left(-\dfrac{18}{7}\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{13}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{59}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{17}{8}\right)\sqrt{45}\right)\right)\right)\\
&=&\left(\left(-\dfrac{138}{7}\right)\sqrt{5}\right)+\left(\left(\left(30\right)\sqrt{5}+\left(-\dfrac{90}{7}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{13}{3}\right)\sqrt{5}\right)+59+\dfrac{125}{2}-\left(\left(-\dfrac{51}{8}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(-\dfrac{138}{7}\right)\sqrt{5}+\left(\left(30\right)\sqrt{5}+\left(-\dfrac{90}{7}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{13}{3}\right)\sqrt{5}\right)+59+\dfrac{125}{2}-\left(\left(-\dfrac{51}{8}\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{2231}{168}\right)\sqrt{5}-\dfrac{243}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{69}{7}\right)\sqrt{20}\right)-\left(\left(\left(6\right)\sqrt{125}+\left(-\dfrac{18}{7}\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{13}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{59}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{17}{8}\right)\sqrt{45}\right)\right)\right)\\
&=&\left(\left(-\dfrac{138}{7}\right)\sqrt{5}\right)-\left(\left(\left(30\right)\sqrt{5}+\left(-\dfrac{90}{7}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{13}{3}\right)\sqrt{5}\right)+59+\dfrac{125}{2}-\left(\left(-\dfrac{51}{8}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(-\dfrac{138}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{1081}{168}\right)\sqrt{5}-\dfrac{243}{2}\right)\\
&=&\left(-\dfrac{138}{7}\right)\sqrt{5}+\left(-\dfrac{1081}{168}\right)\sqrt{5}+\dfrac{243}{2}\\
&=&\left(-\dfrac{4393}{168}\right)\sqrt{5}+\dfrac{243}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{69}{7}\right)\sqrt{20}\right)\times\left(\left(\left(6\right)\sqrt{125}+\left(-\dfrac{18}{7}\right)\sqrt{125}\right)-\left(\left(\left(\dfrac{13}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{59}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{17}{8}\right)\sqrt{45}\right)\right)\right)\\
&=&\left(\left(-\dfrac{138}{7}\right)\sqrt{5}\right)\times\left(\left(\left(30\right)\sqrt{5}+\left(-\dfrac{90}{7}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{13}{3}\right)\sqrt{5}\right)+59+\dfrac{125}{2}-\left(\left(-\dfrac{51}{8}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(-\dfrac{138}{7}\right)\sqrt{5}\right)\left(\left(\dfrac{1081}{168}\right)\sqrt{5}-\dfrac{243}{2}\right)\\
&=&\left(-\dfrac{24863}{196}\right)\sqrt{25}+\left(\dfrac{16767}{7}\right)\sqrt{5}\\
&=&-\dfrac{124315}{196}+\left(\dfrac{16767}{7}\right)\sqrt{5}\\
\end{eqnarray*}