L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{31}{2}\right)\sqrt{175}\right)-\dfrac{53}{7}-\left(\left(-8\right)\sqrt{63}\right)\right)-\left(\left(\left(\dfrac{23}{4}\right)\sqrt{28}\right)-\left(\left(-\dfrac{1}{8}\right)\sqrt{175}\right)-\left(\left(\dfrac{78}{5}\right)\sqrt{175}\right)\right)\) et \( Y=\left(\left(8\right)\sqrt{175}+\left(\dfrac{61}{7}\right)\sqrt{49}+\left(8\right)\sqrt{175}\right)-\left(-\dfrac{7}{2}-\left(\left(1\right)\sqrt{175}\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{21}{5}\right)\sqrt{49}\right)\) . 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(-8\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{31}{2}\right)\sqrt{175}\right)-\dfrac{53}{7}-\left(\left(-8\right)\sqrt{63}\right)\right)-\left(\left(\left(\dfrac{23}{4}\right)\sqrt{28}\right)-\left(\left(-\dfrac{1}{8}\right)\sqrt{175}\right)-\left(\left(\dfrac{78}{5}\right)\sqrt{175}\right)\right)\right)+\left(\left(\left(8\right)\sqrt{175}+\left(\dfrac{61}{7}\right)\sqrt{49}+\left(8\right)\sqrt{175}\right)-\left(-\dfrac{7}{2}-\left(\left(1\right)\sqrt{175}\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{21}{5}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(-24\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{155}{2}\right)\sqrt{7}\right)-\dfrac{53}{7}-\left(\left(-24\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{23}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{5}{8}\right)\sqrt{7}\right)-\left(\left(78\right)\sqrt{7}\right)\right)\right)+\left(\left(\left(40\right)\sqrt{7}+61+\left(40\right)\sqrt{7}\right)-\left(-\dfrac{7}{2}-\left(\left(5\right)\sqrt{7}\right)-\left(\left(-31\right)\sqrt{7}\right)\right)+\dfrac{147}{5}\right)\\
&=&\left(\left(-24\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{155}{2}\right)\sqrt{7}\right)-\dfrac{53}{7}-\left(\left(-24\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{23}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{5}{8}\right)\sqrt{7}\right)-\left(\left(78\right)\sqrt{7}\right)\right)+\left(\left(40\right)\sqrt{7}+61+\left(40\right)\sqrt{7}\right)-\left(-\dfrac{7}{2}-\left(\left(5\right)\sqrt{7}\right)-\left(\left(-31\right)\sqrt{7}\right)\right)+\dfrac{147}{5}\\
&=&\left(\dfrac{1195}{8}\right)\sqrt{7}+\dfrac{7103}{70}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{31}{2}\right)\sqrt{175}\right)-\dfrac{53}{7}-\left(\left(-8\right)\sqrt{63}\right)\right)-\left(\left(\left(\dfrac{23}{4}\right)\sqrt{28}\right)-\left(\left(-\dfrac{1}{8}\right)\sqrt{175}\right)-\left(\left(\dfrac{78}{5}\right)\sqrt{175}\right)\right)\right)-\left(\left(\left(8\right)\sqrt{175}+\left(\dfrac{61}{7}\right)\sqrt{49}+\left(8\right)\sqrt{175}\right)-\left(-\dfrac{7}{2}-\left(\left(1\right)\sqrt{175}\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{21}{5}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(-24\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{155}{2}\right)\sqrt{7}\right)-\dfrac{53}{7}-\left(\left(-24\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{23}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{5}{8}\right)\sqrt{7}\right)-\left(\left(78\right)\sqrt{7}\right)\right)\right)-\left(\left(\left(40\right)\sqrt{7}+61+\left(40\right)\sqrt{7}\right)-\left(-\dfrac{7}{2}-\left(\left(5\right)\sqrt{7}\right)-\left(\left(-31\right)\sqrt{7}\right)\right)+\dfrac{147}{5}\right)\\
&=&\left(\left(\dfrac{763}{8}\right)\sqrt{7}+\dfrac{53}{7}\right)-\left(\left(54\right)\sqrt{7}+\dfrac{939}{10}\right)\\
&=&\left(\dfrac{763}{8}\right)\sqrt{7}+\dfrac{53}{7}+\left(-54\right)\sqrt{7}-\dfrac{939}{10}\\
&=&\left(\dfrac{331}{8}\right)\sqrt{7}-\dfrac{6043}{70}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{31}{2}\right)\sqrt{175}\right)-\dfrac{53}{7}-\left(\left(-8\right)\sqrt{63}\right)\right)-\left(\left(\left(\dfrac{23}{4}\right)\sqrt{28}\right)-\left(\left(-\dfrac{1}{8}\right)\sqrt{175}\right)-\left(\left(\dfrac{78}{5}\right)\sqrt{175}\right)\right)\right)\times\left(\left(\left(8\right)\sqrt{175}+\left(\dfrac{61}{7}\right)\sqrt{49}+\left(8\right)\sqrt{175}\right)-\left(-\dfrac{7}{2}-\left(\left(1\right)\sqrt{175}\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{21}{5}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(-24\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{155}{2}\right)\sqrt{7}\right)-\dfrac{53}{7}-\left(\left(-24\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{23}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{5}{8}\right)\sqrt{7}\right)-\left(\left(78\right)\sqrt{7}\right)\right)\right)\times\left(\left(\left(40\right)\sqrt{7}+61+\left(40\right)\sqrt{7}\right)-\left(-\dfrac{7}{2}-\left(\left(5\right)\sqrt{7}\right)-\left(\left(-31\right)\sqrt{7}\right)\right)+\dfrac{147}{5}\right)\\
&=&\left(\left(\dfrac{763}{8}\right)\sqrt{7}+\dfrac{53}{7}\right)\left(\left(54\right)\sqrt{7}+\dfrac{939}{10}\right)\\
&=&\left(\dfrac{20601}{4}\right)\sqrt{49}+\left(\dfrac{5244159}{560}\right)\sqrt{7}+\dfrac{49767}{70}\\
&=&\dfrac{5146779}{140}+\left(\dfrac{5244159}{560}\right)\sqrt{7}\\
\end{eqnarray*}