L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{39}{2}\right)\sqrt{28}\right)-\left(\left(\dfrac{68}{7}\right)\sqrt{28}\right)+\dfrac{23}{8}\) et \( Y=\left(\left(3\right)\sqrt{63}+\left(-\dfrac{33}{2}\right)\sqrt{49}+\left(-6\right)\sqrt{49}\right)-\left(1+\left(-\dfrac{47}{4}\right)\sqrt{63}+\left(-7\right)\sqrt{28}\right)-\left(\left(-\dfrac{20}{9}\right)\sqrt{49}+\left(3\right)\sqrt{175}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\dfrac{39}{2}\right)\sqrt{28}\right)-\left(\left(\dfrac{68}{7}\right)\sqrt{28}\right)+\dfrac{23}{8}\right)+\left(\left(\left(3\right)\sqrt{63}+\left(-\dfrac{33}{2}\right)\sqrt{49}+\left(-6\right)\sqrt{49}\right)-\left(1+\left(-\dfrac{47}{4}\right)\sqrt{63}+\left(-7\right)\sqrt{28}\right)-\left(\left(-\dfrac{20}{9}\right)\sqrt{49}+\left(3\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(39\right)\sqrt{7}\right)-\left(\left(\dfrac{136}{7}\right)\sqrt{7}\right)+\dfrac{23}{8}\right)+\left(\left(\left(9\right)\sqrt{7}-\dfrac{231}{2}-42\right)-\left(1+\left(-\dfrac{141}{4}\right)\sqrt{7}+\left(-14\right)\sqrt{7}\right)-\left(-\dfrac{140}{9}+\left(15\right)\sqrt{7}\right)\right)\\
&=&\left(\left(39\right)\sqrt{7}\right)-\left(\left(\dfrac{136}{7}\right)\sqrt{7}\right)+\dfrac{23}{8}+\left(\left(9\right)\sqrt{7}-\dfrac{231}{2}-42\right)-\left(1+\left(-\dfrac{141}{4}\right)\sqrt{7}+\left(-14\right)\sqrt{7}\right)-\left(-\dfrac{140}{9}+\left(15\right)\sqrt{7}\right)\\
&=&\left(\dfrac{1759}{28}\right)\sqrt{7}-\dfrac{10085}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{39}{2}\right)\sqrt{28}\right)-\left(\left(\dfrac{68}{7}\right)\sqrt{28}\right)+\dfrac{23}{8}\right)-\left(\left(\left(3\right)\sqrt{63}+\left(-\dfrac{33}{2}\right)\sqrt{49}+\left(-6\right)\sqrt{49}\right)-\left(1+\left(-\dfrac{47}{4}\right)\sqrt{63}+\left(-7\right)\sqrt{28}\right)-\left(\left(-\dfrac{20}{9}\right)\sqrt{49}+\left(3\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(39\right)\sqrt{7}\right)-\left(\left(\dfrac{136}{7}\right)\sqrt{7}\right)+\dfrac{23}{8}\right)-\left(\left(\left(9\right)\sqrt{7}-\dfrac{231}{2}-42\right)-\left(1+\left(-\dfrac{141}{4}\right)\sqrt{7}+\left(-14\right)\sqrt{7}\right)-\left(-\dfrac{140}{9}+\left(15\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{137}{7}\right)\sqrt{7}+\dfrac{23}{8}\right)-\left(\left(\dfrac{173}{4}\right)\sqrt{7}-\dfrac{2573}{18}\right)\\
&=&\left(\dfrac{137}{7}\right)\sqrt{7}+\dfrac{23}{8}+\left(-\dfrac{173}{4}\right)\sqrt{7}+\dfrac{2573}{18}\\
&=&\left(-\dfrac{663}{28}\right)\sqrt{7}+\dfrac{10499}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{39}{2}\right)\sqrt{28}\right)-\left(\left(\dfrac{68}{7}\right)\sqrt{28}\right)+\dfrac{23}{8}\right)\times\left(\left(\left(3\right)\sqrt{63}+\left(-\dfrac{33}{2}\right)\sqrt{49}+\left(-6\right)\sqrt{49}\right)-\left(1+\left(-\dfrac{47}{4}\right)\sqrt{63}+\left(-7\right)\sqrt{28}\right)-\left(\left(-\dfrac{20}{9}\right)\sqrt{49}+\left(3\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(39\right)\sqrt{7}\right)-\left(\left(\dfrac{136}{7}\right)\sqrt{7}\right)+\dfrac{23}{8}\right)\times\left(\left(\left(9\right)\sqrt{7}-\dfrac{231}{2}-42\right)-\left(1+\left(-\dfrac{141}{4}\right)\sqrt{7}+\left(-14\right)\sqrt{7}\right)-\left(-\dfrac{140}{9}+\left(15\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{137}{7}\right)\sqrt{7}+\dfrac{23}{8}\right)\left(\left(\dfrac{173}{4}\right)\sqrt{7}-\dfrac{2573}{18}\right)\\
&=&\left(\dfrac{23701}{28}\right)\sqrt{49}+\left(-\dfrac{5389339}{2016}\right)\sqrt{7}-\dfrac{59179}{144}\\
&=&\dfrac{794057}{144}+\left(-\dfrac{5389339}{2016}\right)\sqrt{7}\\
\end{eqnarray*}