L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=0-\left(-\dfrac{14}{5}-\left(\left(-\dfrac{61}{6}\right)\sqrt{28}\right)\right)-\left(-9+\left(-4\right)\sqrt{175}+\left(\dfrac{2}{3}\right)\sqrt{49}\right)\) et \( Y=\left(\left(-\dfrac{26}{5}\right)\sqrt{28}+\left(1\right)\sqrt{49}+\dfrac{1}{3}+\left(-\dfrac{61}{8}\right)\sqrt{49}\right)+\dfrac{53}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{49}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(0-\left(-\dfrac{14}{5}-\left(\left(-\dfrac{61}{6}\right)\sqrt{28}\right)\right)-\left(-9+\left(-4\right)\sqrt{175}+\left(\dfrac{2}{3}\right)\sqrt{49}\right)\right)+\left(\left(\left(-\dfrac{26}{5}\right)\sqrt{28}+\left(1\right)\sqrt{49}+\dfrac{1}{3}+\left(-\dfrac{61}{8}\right)\sqrt{49}\right)+\dfrac{53}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{49}\right)\right)\\
&=&\left(0-\left(-\dfrac{14}{5}-\left(\left(-\dfrac{61}{3}\right)\sqrt{7}\right)\right)-\left(-9+\left(-20\right)\sqrt{7}+\dfrac{14}{3}\right)\right)+\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{7}+7+\dfrac{1}{3}-\dfrac{427}{8}\right)+\dfrac{53}{3}-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}\right)-7\right)\\
&=&0-\left(-\dfrac{14}{5}-\left(\left(-\dfrac{61}{3}\right)\sqrt{7}\right)\right)-\left(-9+\left(-20\right)\sqrt{7}+\dfrac{14}{3}\right)+\left(\left(-\dfrac{52}{5}\right)\sqrt{7}+7+\dfrac{1}{3}-\dfrac{427}{8}\right)+\dfrac{53}{3}-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}\right)-7\\
&=&-\dfrac{3389}{120}+\left(\dfrac{173}{30}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(0-\left(-\dfrac{14}{5}-\left(\left(-\dfrac{61}{6}\right)\sqrt{28}\right)\right)-\left(-9+\left(-4\right)\sqrt{175}+\left(\dfrac{2}{3}\right)\sqrt{49}\right)\right)-\left(\left(\left(-\dfrac{26}{5}\right)\sqrt{28}+\left(1\right)\sqrt{49}+\dfrac{1}{3}+\left(-\dfrac{61}{8}\right)\sqrt{49}\right)+\dfrac{53}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{49}\right)\right)\\
&=&\left(0-\left(-\dfrac{14}{5}-\left(\left(-\dfrac{61}{3}\right)\sqrt{7}\right)\right)-\left(-9+\left(-20\right)\sqrt{7}+\dfrac{14}{3}\right)\right)-\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{7}+7+\dfrac{1}{3}-\dfrac{427}{8}\right)+\dfrac{53}{3}-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}\right)-7\right)\\
&=&\left(\dfrac{107}{15}+\left(-\dfrac{1}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{61}{10}\right)\sqrt{7}-\dfrac{283}{8}\right)\\
&=&\dfrac{107}{15}+\left(-\dfrac{1}{3}\right)\sqrt{7}+\left(-\dfrac{61}{10}\right)\sqrt{7}+\dfrac{283}{8}\\
&=&\dfrac{5101}{120}+\left(-\dfrac{193}{30}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(0-\left(-\dfrac{14}{5}-\left(\left(-\dfrac{61}{6}\right)\sqrt{28}\right)\right)-\left(-9+\left(-4\right)\sqrt{175}+\left(\dfrac{2}{3}\right)\sqrt{49}\right)\right)\times\left(\left(\left(-\dfrac{26}{5}\right)\sqrt{28}+\left(1\right)\sqrt{49}+\dfrac{1}{3}+\left(-\dfrac{61}{8}\right)\sqrt{49}\right)+\dfrac{53}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{49}\right)\right)\\
&=&\left(0-\left(-\dfrac{14}{5}-\left(\left(-\dfrac{61}{3}\right)\sqrt{7}\right)\right)-\left(-9+\left(-20\right)\sqrt{7}+\dfrac{14}{3}\right)\right)\times\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{7}+7+\dfrac{1}{3}-\dfrac{427}{8}\right)+\dfrac{53}{3}-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}\right)-7\right)\\
&=&\left(\dfrac{107}{15}+\left(-\dfrac{1}{3}\right)\sqrt{7}\right)\left(\left(\dfrac{61}{10}\right)\sqrt{7}-\dfrac{283}{8}\right)\\
&=&\left(\dfrac{11061}{200}\right)\sqrt{7}-\dfrac{30281}{120}+\left(-\dfrac{61}{30}\right)\sqrt{49}\\
&=&\left(\dfrac{11061}{200}\right)\sqrt{7}-\dfrac{10663}{40}\\
\end{eqnarray*}