L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{5}{2}\right)\sqrt{28}\) et \( Y=\left(\dfrac{25}{2}\right)\sqrt{63}+\left(\dfrac{17}{5}\right)\sqrt{49}+\dfrac{22}{9}+\left(-\dfrac{77}{3}\right)\sqrt{63}+\left(-\dfrac{76}{7}\right)\sqrt{175}+\left(-\dfrac{57}{4}\right)\sqrt{28}+\left(-1\right)\sqrt{49}+\left(\dfrac{59}{8}\right)\sqrt{49}+4+\left(-\dfrac{25}{4}\right)\sqrt{49}+\left(-\dfrac{55}{8}\right)\sqrt{49}+\dfrac{63}{4}+\left(\dfrac{23}{2}\right)\sqrt{28}+0\) . 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{5}{2}\right)\sqrt{28}\right)+\left(\left(\dfrac{25}{2}\right)\sqrt{63}+\left(\dfrac{17}{5}\right)\sqrt{49}+\dfrac{22}{9}+\left(-\dfrac{77}{3}\right)\sqrt{63}+\left(-\dfrac{76}{7}\right)\sqrt{175}+\left(-\dfrac{57}{4}\right)\sqrt{28}+\left(-1\right)\sqrt{49}+\left(\dfrac{59}{8}\right)\sqrt{49}+4+\left(-\dfrac{25}{4}\right)\sqrt{49}+\left(-\dfrac{55}{8}\right)\sqrt{49}+\dfrac{63}{4}+\left(\dfrac{23}{2}\right)\sqrt{28}+0\right)\\
&=&\left(\left(-5\right)\sqrt{7}\right)+\left(\left(\dfrac{75}{2}\right)\sqrt{7}+\dfrac{119}{5}+\dfrac{22}{9}+\left(-77\right)\sqrt{7}+\left(-\dfrac{380}{7}\right)\sqrt{7}+\left(-\dfrac{57}{2}\right)\sqrt{7}-7+\dfrac{413}{8}+4-\dfrac{175}{4}-\dfrac{385}{8}+\dfrac{63}{4}+\left(23\right)\sqrt{7}+0\right)\\
&=&\left(-5\right)\sqrt{7}+\left(\dfrac{75}{2}\right)\sqrt{7}+\dfrac{119}{5}+\dfrac{22}{9}+\left(-77\right)\sqrt{7}+\left(-\dfrac{380}{7}\right)\sqrt{7}+\left(-\dfrac{57}{2}\right)\sqrt{7}-7+\dfrac{413}{8}+4-\dfrac{175}{4}-\dfrac{385}{8}+\dfrac{63}{4}+\left(23\right)\sqrt{7}+0\\
&=&\left(-\dfrac{730}{7}\right)\sqrt{7}-\dfrac{113}{90}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{5}{2}\right)\sqrt{28}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{63}+\left(\dfrac{17}{5}\right)\sqrt{49}+\dfrac{22}{9}+\left(-\dfrac{77}{3}\right)\sqrt{63}+\left(-\dfrac{76}{7}\right)\sqrt{175}+\left(-\dfrac{57}{4}\right)\sqrt{28}+\left(-1\right)\sqrt{49}+\left(\dfrac{59}{8}\right)\sqrt{49}+4+\left(-\dfrac{25}{4}\right)\sqrt{49}+\left(-\dfrac{55}{8}\right)\sqrt{49}+\dfrac{63}{4}+\left(\dfrac{23}{2}\right)\sqrt{28}+0\right)\\
&=&\left(\left(-5\right)\sqrt{7}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{7}+\dfrac{119}{5}+\dfrac{22}{9}+\left(-77\right)\sqrt{7}+\left(-\dfrac{380}{7}\right)\sqrt{7}+\left(-\dfrac{57}{2}\right)\sqrt{7}-7+\dfrac{413}{8}+4-\dfrac{175}{4}-\dfrac{385}{8}+\dfrac{63}{4}+\left(23\right)\sqrt{7}+0\right)\\
&=&\left(\left(-5\right)\sqrt{7}\right)-\left(\left(-\dfrac{695}{7}\right)\sqrt{7}-\dfrac{113}{90}\right)\\
&=&\left(-5\right)\sqrt{7}+\left(\dfrac{695}{7}\right)\sqrt{7}+\dfrac{113}{90}\\
&=&\left(\dfrac{660}{7}\right)\sqrt{7}+\dfrac{113}{90}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{5}{2}\right)\sqrt{28}\right)\times\left(\left(\dfrac{25}{2}\right)\sqrt{63}+\left(\dfrac{17}{5}\right)\sqrt{49}+\dfrac{22}{9}+\left(-\dfrac{77}{3}\right)\sqrt{63}+\left(-\dfrac{76}{7}\right)\sqrt{175}+\left(-\dfrac{57}{4}\right)\sqrt{28}+\left(-1\right)\sqrt{49}+\left(\dfrac{59}{8}\right)\sqrt{49}+4+\left(-\dfrac{25}{4}\right)\sqrt{49}+\left(-\dfrac{55}{8}\right)\sqrt{49}+\dfrac{63}{4}+\left(\dfrac{23}{2}\right)\sqrt{28}+0\right)\\
&=&\left(\left(-5\right)\sqrt{7}\right)\times\left(\left(\dfrac{75}{2}\right)\sqrt{7}+\dfrac{119}{5}+\dfrac{22}{9}+\left(-77\right)\sqrt{7}+\left(-\dfrac{380}{7}\right)\sqrt{7}+\left(-\dfrac{57}{2}\right)\sqrt{7}-7+\dfrac{413}{8}+4-\dfrac{175}{4}-\dfrac{385}{8}+\dfrac{63}{4}+\left(23\right)\sqrt{7}+0\right)\\
&=&\left(\left(-5\right)\sqrt{7}\right)\left(\left(-\dfrac{695}{7}\right)\sqrt{7}-\dfrac{113}{90}\right)\\
&=&\left(\dfrac{3475}{7}\right)\sqrt{49}+\left(\dfrac{113}{18}\right)\sqrt{7}\\
&=&3475+\left(\dfrac{113}{18}\right)\sqrt{7}\\
\end{eqnarray*}