L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=0-\left(\left(-\dfrac{43}{4}\right)\sqrt{175}+\left(\dfrac{31}{2}\right)\sqrt{49}+\left(\dfrac{2}{3}\right)\sqrt{28}\right)\) et \( Y=\left(\dfrac{31}{5}\right)\sqrt{63}+\left(-\dfrac{11}{3}\right)\sqrt{63}+\left(2\right)\sqrt{63}+\left(\left(-\dfrac{19}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{55}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{41}{5}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{49}\right)-\left(\left(\dfrac{4}{5}\right)\sqrt{63}\right)+\left(6\right)\sqrt{63}+\left(\dfrac{11}{2}\right)\sqrt{63}+\left(\dfrac{38}{3}\right)\sqrt{63}\) . 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(\left(-\dfrac{43}{4}\right)\sqrt{175}+\left(\dfrac{31}{2}\right)\sqrt{49}+\left(\dfrac{2}{3}\right)\sqrt{28}\right)\right)+\left(\left(\dfrac{31}{5}\right)\sqrt{63}+\left(-\dfrac{11}{3}\right)\sqrt{63}+\left(2\right)\sqrt{63}+\left(\left(-\dfrac{19}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{55}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{41}{5}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{49}\right)-\left(\left(\dfrac{4}{5}\right)\sqrt{63}\right)+\left(6\right)\sqrt{63}+\left(\dfrac{11}{2}\right)\sqrt{63}+\left(\dfrac{38}{3}\right)\sqrt{63}\right)\\
&=&\left(0-\left(\left(-\dfrac{215}{4}\right)\sqrt{7}+\dfrac{217}{2}+\left(\dfrac{4}{3}\right)\sqrt{7}\right)\right)+\left(\left(\dfrac{93}{5}\right)\sqrt{7}+\left(-11\right)\sqrt{7}+\left(6\right)\sqrt{7}-\dfrac{133}{4}-\left(\left(\dfrac{55}{3}\right)\sqrt{7}\right)+\dfrac{287}{5}-0-\left(\left(\dfrac{12}{5}\right)\sqrt{7}\right)+\left(18\right)\sqrt{7}+\left(\dfrac{33}{2}\right)\sqrt{7}+\left(38\right)\sqrt{7}\right)\\
&=&0-\left(\left(-\dfrac{215}{4}\right)\sqrt{7}+\dfrac{217}{2}+\left(\dfrac{4}{3}\right)\sqrt{7}\right)+\left(\dfrac{93}{5}\right)\sqrt{7}+\left(-11\right)\sqrt{7}+\left(6\right)\sqrt{7}-\dfrac{133}{4}-\left(\left(\dfrac{55}{3}\right)\sqrt{7}\right)+\dfrac{287}{5}-0-\left(\left(\dfrac{12}{5}\right)\sqrt{7}\right)+\left(18\right)\sqrt{7}+\left(\dfrac{33}{2}\right)\sqrt{7}+\left(38\right)\sqrt{7}\\
&=&-\dfrac{1687}{20}+\left(\dfrac{7067}{60}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(0-\left(\left(-\dfrac{43}{4}\right)\sqrt{175}+\left(\dfrac{31}{2}\right)\sqrt{49}+\left(\dfrac{2}{3}\right)\sqrt{28}\right)\right)-\left(\left(\dfrac{31}{5}\right)\sqrt{63}+\left(-\dfrac{11}{3}\right)\sqrt{63}+\left(2\right)\sqrt{63}+\left(\left(-\dfrac{19}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{55}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{41}{5}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{49}\right)-\left(\left(\dfrac{4}{5}\right)\sqrt{63}\right)+\left(6\right)\sqrt{63}+\left(\dfrac{11}{2}\right)\sqrt{63}+\left(\dfrac{38}{3}\right)\sqrt{63}\right)\\
&=&\left(0-\left(\left(-\dfrac{215}{4}\right)\sqrt{7}+\dfrac{217}{2}+\left(\dfrac{4}{3}\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{93}{5}\right)\sqrt{7}+\left(-11\right)\sqrt{7}+\left(6\right)\sqrt{7}-\dfrac{133}{4}-\left(\left(\dfrac{55}{3}\right)\sqrt{7}\right)+\dfrac{287}{5}-0-\left(\left(\dfrac{12}{5}\right)\sqrt{7}\right)+\left(18\right)\sqrt{7}+\left(\dfrac{33}{2}\right)\sqrt{7}+\left(38\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{217}{2}+\left(\dfrac{629}{12}\right)\sqrt{7}\right)-\left(\left(\dfrac{1961}{30}\right)\sqrt{7}+\dfrac{483}{20}\right)\\
&=&-\dfrac{217}{2}+\left(\dfrac{629}{12}\right)\sqrt{7}+\left(-\dfrac{1961}{30}\right)\sqrt{7}-\dfrac{483}{20}\\
&=&-\dfrac{2653}{20}+\left(-\dfrac{259}{20}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(0-\left(\left(-\dfrac{43}{4}\right)\sqrt{175}+\left(\dfrac{31}{2}\right)\sqrt{49}+\left(\dfrac{2}{3}\right)\sqrt{28}\right)\right)\times\left(\left(\dfrac{31}{5}\right)\sqrt{63}+\left(-\dfrac{11}{3}\right)\sqrt{63}+\left(2\right)\sqrt{63}+\left(\left(-\dfrac{19}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{55}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{41}{5}\right)\sqrt{49}\right)-\left(\left(0\right)\sqrt{49}\right)-\left(\left(\dfrac{4}{5}\right)\sqrt{63}\right)+\left(6\right)\sqrt{63}+\left(\dfrac{11}{2}\right)\sqrt{63}+\left(\dfrac{38}{3}\right)\sqrt{63}\right)\\
&=&\left(0-\left(\left(-\dfrac{215}{4}\right)\sqrt{7}+\dfrac{217}{2}+\left(\dfrac{4}{3}\right)\sqrt{7}\right)\right)\times\left(\left(\dfrac{93}{5}\right)\sqrt{7}+\left(-11\right)\sqrt{7}+\left(6\right)\sqrt{7}-\dfrac{133}{4}-\left(\left(\dfrac{55}{3}\right)\sqrt{7}\right)+\dfrac{287}{5}-0-\left(\left(\dfrac{12}{5}\right)\sqrt{7}\right)+\left(18\right)\sqrt{7}+\left(\dfrac{33}{2}\right)\sqrt{7}+\left(38\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{217}{2}+\left(\dfrac{629}{12}\right)\sqrt{7}\right)\left(\left(\dfrac{1961}{30}\right)\sqrt{7}+\dfrac{483}{20}\right)\\
&=&\left(-\dfrac{1398341}{240}\right)\sqrt{7}-\dfrac{104811}{40}+\left(\dfrac{1233469}{360}\right)\sqrt{49}\\
&=&\left(-\dfrac{1398341}{240}\right)\sqrt{7}+\dfrac{961373}{45}\\
\end{eqnarray*}