L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-7\right)\sqrt{175}+\left(\left(-\dfrac{1}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{2}\right)\sqrt{63}\right)-\left(\left(6\right)\sqrt{63}\right)+\left(-\dfrac{5}{3}\right)\sqrt{49}+\left(\dfrac{73}{7}\right)\sqrt{175}+\left(0\right)\sqrt{49}\) et \( Y=\left(\left(-2\right)\sqrt{49}\right)-\left(\left(\left(9\right)\sqrt{175}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{63}\right)+2\right)-\left(-\dfrac{71}{7}-5+\left(\dfrac{53}{3}\right)\sqrt{49}-2\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-7\right)\sqrt{175}+\left(\left(-\dfrac{1}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{2}\right)\sqrt{63}\right)-\left(\left(6\right)\sqrt{63}\right)+\left(-\dfrac{5}{3}\right)\sqrt{49}+\left(\dfrac{73}{7}\right)\sqrt{175}+\left(0\right)\sqrt{49}\right)+\left(\left(\left(-2\right)\sqrt{49}\right)-\left(\left(\left(9\right)\sqrt{175}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{63}\right)+2\right)-\left(-\dfrac{71}{7}-5+\left(\dfrac{53}{3}\right)\sqrt{49}-2\right)\right)\\
&=&\left(\left(-35\right)\sqrt{7}+\left(\left(-\dfrac{2}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{201}{2}\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\dfrac{35}{3}+\left(\dfrac{365}{7}\right)\sqrt{7}+0\right)+\left(-14-\left(\left(\left(45\right)\sqrt{7}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{7}\right)+2\right)-\left(-\dfrac{71}{7}-5+\dfrac{371}{3}-2\right)\right)\\
&=&\left(-35\right)\sqrt{7}+\left(\left(-\dfrac{2}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{201}{2}\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\dfrac{35}{3}+\left(\dfrac{365}{7}\right)\sqrt{7}+0-14-\left(\left(\left(45\right)\sqrt{7}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{7}\right)+2\right)-\left(-\dfrac{71}{7}-5+\dfrac{371}{3}-2\right)\\
&=&\left(\dfrac{13193}{280}\right)\sqrt{7}-\dfrac{2818}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-7\right)\sqrt{175}+\left(\left(-\dfrac{1}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{2}\right)\sqrt{63}\right)-\left(\left(6\right)\sqrt{63}\right)+\left(-\dfrac{5}{3}\right)\sqrt{49}+\left(\dfrac{73}{7}\right)\sqrt{175}+\left(0\right)\sqrt{49}\right)-\left(\left(\left(-2\right)\sqrt{49}\right)-\left(\left(\left(9\right)\sqrt{175}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{63}\right)+2\right)-\left(-\dfrac{71}{7}-5+\left(\dfrac{53}{3}\right)\sqrt{49}-2\right)\right)\\
&=&\left(\left(-35\right)\sqrt{7}+\left(\left(-\dfrac{2}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{201}{2}\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\dfrac{35}{3}+\left(\dfrac{365}{7}\right)\sqrt{7}+0\right)-\left(-14-\left(\left(\left(45\right)\sqrt{7}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{7}\right)+2\right)-\left(-\dfrac{71}{7}-5+\dfrac{371}{3}-2\right)\right)\\
&=&\left(\left(\dfrac{6947}{70}\right)\sqrt{7}-\dfrac{35}{3}\right)-\left(-\dfrac{2573}{21}+\left(-\dfrac{417}{8}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{6947}{70}\right)\sqrt{7}-\dfrac{35}{3}+\dfrac{2573}{21}+\left(\dfrac{417}{8}\right)\sqrt{7}\\
&=&\left(\dfrac{42383}{280}\right)\sqrt{7}+\dfrac{776}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-7\right)\sqrt{175}+\left(\left(-\dfrac{1}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{2}\right)\sqrt{63}\right)-\left(\left(6\right)\sqrt{63}\right)+\left(-\dfrac{5}{3}\right)\sqrt{49}+\left(\dfrac{73}{7}\right)\sqrt{175}+\left(0\right)\sqrt{49}\right)\times\left(\left(\left(-2\right)\sqrt{49}\right)-\left(\left(\left(9\right)\sqrt{175}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{63}\right)+2\right)-\left(-\dfrac{71}{7}-5+\left(\dfrac{53}{3}\right)\sqrt{49}-2\right)\right)\\
&=&\left(\left(-35\right)\sqrt{7}+\left(\left(-\dfrac{2}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{201}{2}\right)\sqrt{7}\right)-\left(\left(18\right)\sqrt{7}\right)-\dfrac{35}{3}+\left(\dfrac{365}{7}\right)\sqrt{7}+0\right)\times\left(-14-\left(\left(\left(45\right)\sqrt{7}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{7}\right)+2\right)-\left(-\dfrac{71}{7}-5+\dfrac{371}{3}-2\right)\right)\\
&=&\left(\left(\dfrac{6947}{70}\right)\sqrt{7}-\dfrac{35}{3}\right)\left(-\dfrac{2573}{21}+\left(-\dfrac{417}{8}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{67922749}{5880}\right)\sqrt{7}+\left(-\dfrac{2896899}{560}\right)\sqrt{49}+\dfrac{12865}{9}\\
&=&\left(-\dfrac{67922749}{5880}\right)\sqrt{7}-\dfrac{25042891}{720}\\
\end{eqnarray*}