L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{24}{5}\right)\sqrt{28}\right)-\left(\left(\left(2\right)\sqrt{63}\right)-\left(\left(9\right)\sqrt{28}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{63}\right)\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{28}\right)\) et \( Y=\left(-2\right)\sqrt{28}+\left(-\dfrac{11}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(-3\right)\sqrt{63}+\left(\left(5\right)\sqrt{175}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{175}\right)+\left(\left(-\dfrac{8}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{12}{5}\right)\sqrt{49}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{24}{5}\right)\sqrt{28}\right)-\left(\left(\left(2\right)\sqrt{63}\right)-\left(\left(9\right)\sqrt{28}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{63}\right)\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{28}\right)\right)+\left(\left(-2\right)\sqrt{28}+\left(-\dfrac{11}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(-3\right)\sqrt{63}+\left(\left(5\right)\sqrt{175}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{175}\right)+\left(\left(-\dfrac{8}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{12}{5}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(-\dfrac{48}{5}\right)\sqrt{7}\right)-\left(\left(\left(6\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\left(\left(-5\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{59}{4}\right)\sqrt{7}\right)\right)+\left(\left(-4\right)\sqrt{7}+\left(-\dfrac{55}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}+\left(-9\right)\sqrt{7}+\left(\left(25\right)\sqrt{7}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{7}\right)-\left(\left(77\right)\sqrt{7}\right)+\left(\left(-\dfrac{40}{7}\right)\sqrt{7}\right)+\dfrac{84}{5}\right)\\
&=&\left(\left(-\dfrac{48}{5}\right)\sqrt{7}\right)-\left(\left(\left(6\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\left(\left(-5\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{59}{4}\right)\sqrt{7}\right)+\left(-4\right)\sqrt{7}+\left(-\dfrac{55}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}+\left(-9\right)\sqrt{7}+\left(\left(25\right)\sqrt{7}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{7}\right)-\left(\left(77\right)\sqrt{7}\right)+\left(\left(-\dfrac{40}{7}\right)\sqrt{7}\right)+\dfrac{84}{5}\\
&=&\left(-\dfrac{54697}{420}\right)\sqrt{7}+\dfrac{84}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{24}{5}\right)\sqrt{28}\right)-\left(\left(\left(2\right)\sqrt{63}\right)-\left(\left(9\right)\sqrt{28}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{63}\right)\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{28}\right)\right)-\left(\left(-2\right)\sqrt{28}+\left(-\dfrac{11}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(-3\right)\sqrt{63}+\left(\left(5\right)\sqrt{175}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{175}\right)+\left(\left(-\dfrac{8}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{12}{5}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(-\dfrac{48}{5}\right)\sqrt{7}\right)-\left(\left(\left(6\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\left(\left(-5\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{59}{4}\right)\sqrt{7}\right)\right)-\left(\left(-4\right)\sqrt{7}+\left(-\dfrac{55}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}+\left(-9\right)\sqrt{7}+\left(\left(25\right)\sqrt{7}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{7}\right)-\left(\left(77\right)\sqrt{7}\right)+\left(\left(-\dfrac{40}{7}\right)\sqrt{7}\right)+\dfrac{84}{5}\right)\\
&=&\left(\left(-\dfrac{347}{20}\right)\sqrt{7}\right)-\left(\left(-\dfrac{4741}{42}\right)\sqrt{7}+\dfrac{84}{5}\right)\\
&=&\left(-\dfrac{347}{20}\right)\sqrt{7}+\left(\dfrac{4741}{42}\right)\sqrt{7}-\dfrac{84}{5}\\
&=&\left(\dfrac{40123}{420}\right)\sqrt{7}-\dfrac{84}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{24}{5}\right)\sqrt{28}\right)-\left(\left(\left(2\right)\sqrt{63}\right)-\left(\left(9\right)\sqrt{28}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{63}\right)\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{28}\right)\right)\times\left(\left(-2\right)\sqrt{28}+\left(-\dfrac{11}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(-3\right)\sqrt{63}+\left(\left(5\right)\sqrt{175}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{175}\right)+\left(\left(-\dfrac{8}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{12}{5}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(-\dfrac{48}{5}\right)\sqrt{7}\right)-\left(\left(\left(6\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\left(\left(-5\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{59}{4}\right)\sqrt{7}\right)\right)\times\left(\left(-4\right)\sqrt{7}+\left(-\dfrac{55}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}+\left(-9\right)\sqrt{7}+\left(\left(25\right)\sqrt{7}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{7}\right)-\left(\left(77\right)\sqrt{7}\right)+\left(\left(-\dfrac{40}{7}\right)\sqrt{7}\right)+\dfrac{84}{5}\right)\\
&=&\left(\left(-\dfrac{347}{20}\right)\sqrt{7}\right)\left(\left(-\dfrac{4741}{42}\right)\sqrt{7}+\dfrac{84}{5}\right)\\
&=&\left(\dfrac{1645127}{840}\right)\sqrt{49}+\left(-\dfrac{7287}{25}\right)\sqrt{7}\\
&=&\dfrac{1645127}{120}+\left(-\dfrac{7287}{25}\right)\sqrt{7}\\
\end{eqnarray*}