L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-9\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{28}\right)-\dfrac{61}{7}+\left(-9\right)\sqrt{49}+\left(\dfrac{73}{2}\right)\sqrt{63}+\left(-\dfrac{3}{2}\right)\sqrt{175}+\left(-\dfrac{17}{7}\right)\sqrt{49}+\left(-\dfrac{17}{7}\right)\sqrt{49}-\dfrac{50}{9}+\left(-2\right)\sqrt{175}+\dfrac{15}{4}+\left(-\dfrac{74}{3}\right)\sqrt{175}-1+\left(\dfrac{36}{7}\right)\sqrt{175}\) et \( Y=\left(\dfrac{19}{4}\right)\sqrt{63}+\left(\left(\dfrac{38}{3}\right)\sqrt{28}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{175}\right)\) . 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(-9\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{28}\right)-\dfrac{61}{7}+\left(-9\right)\sqrt{49}+\left(\dfrac{73}{2}\right)\sqrt{63}+\left(-\dfrac{3}{2}\right)\sqrt{175}+\left(-\dfrac{17}{7}\right)\sqrt{49}+\left(-\dfrac{17}{7}\right)\sqrt{49}-\dfrac{50}{9}+\left(-2\right)\sqrt{175}+\dfrac{15}{4}+\left(-\dfrac{74}{3}\right)\sqrt{175}-1+\left(\dfrac{36}{7}\right)\sqrt{175}\right)+\left(\left(\dfrac{19}{4}\right)\sqrt{63}+\left(\left(\dfrac{38}{3}\right)\sqrt{28}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{175}\right)\right)\\
&=&\left(-63-\left(\left(19\right)\sqrt{7}\right)-\dfrac{61}{7}-63+\left(\dfrac{219}{2}\right)\sqrt{7}+\left(-\dfrac{15}{2}\right)\sqrt{7}-17-17-\dfrac{50}{9}+\left(-10\right)\sqrt{7}+\dfrac{15}{4}+\left(-\dfrac{370}{3}\right)\sqrt{7}-1+\left(\dfrac{180}{7}\right)\sqrt{7}\right)+\left(\left(\dfrac{57}{4}\right)\sqrt{7}+\left(\left(\dfrac{76}{3}\right)\sqrt{7}\right)+\dfrac{245}{8}-\left(\left(20\right)\sqrt{7}\right)\right)\\
&=&-63-\left(\left(19\right)\sqrt{7}\right)-\dfrac{61}{7}-63+\left(\dfrac{219}{2}\right)\sqrt{7}+\left(-\dfrac{15}{2}\right)\sqrt{7}-17-17-\dfrac{50}{9}+\left(-10\right)\sqrt{7}+\dfrac{15}{4}+\left(-\dfrac{370}{3}\right)\sqrt{7}-1+\left(\dfrac{180}{7}\right)\sqrt{7}+\left(\dfrac{57}{4}\right)\sqrt{7}+\left(\left(\dfrac{76}{3}\right)\sqrt{7}\right)+\dfrac{245}{8}-\left(\left(20\right)\sqrt{7}\right)\\
&=&-\dfrac{71011}{504}+\left(-\dfrac{141}{28}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-9\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{28}\right)-\dfrac{61}{7}+\left(-9\right)\sqrt{49}+\left(\dfrac{73}{2}\right)\sqrt{63}+\left(-\dfrac{3}{2}\right)\sqrt{175}+\left(-\dfrac{17}{7}\right)\sqrt{49}+\left(-\dfrac{17}{7}\right)\sqrt{49}-\dfrac{50}{9}+\left(-2\right)\sqrt{175}+\dfrac{15}{4}+\left(-\dfrac{74}{3}\right)\sqrt{175}-1+\left(\dfrac{36}{7}\right)\sqrt{175}\right)-\left(\left(\dfrac{19}{4}\right)\sqrt{63}+\left(\left(\dfrac{38}{3}\right)\sqrt{28}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{175}\right)\right)\\
&=&\left(-63-\left(\left(19\right)\sqrt{7}\right)-\dfrac{61}{7}-63+\left(\dfrac{219}{2}\right)\sqrt{7}+\left(-\dfrac{15}{2}\right)\sqrt{7}-17-17-\dfrac{50}{9}+\left(-10\right)\sqrt{7}+\dfrac{15}{4}+\left(-\dfrac{370}{3}\right)\sqrt{7}-1+\left(\dfrac{180}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{7}+\left(\left(\dfrac{76}{3}\right)\sqrt{7}\right)+\dfrac{245}{8}-\left(\left(20\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{43223}{252}+\left(-\dfrac{517}{21}\right)\sqrt{7}\right)-\left(\left(\dfrac{235}{12}\right)\sqrt{7}+\dfrac{245}{8}\right)\\
&=&-\dfrac{43223}{252}+\left(-\dfrac{517}{21}\right)\sqrt{7}+\left(-\dfrac{235}{12}\right)\sqrt{7}-\dfrac{245}{8}\\
&=&-\dfrac{101881}{504}+\left(-\dfrac{3713}{84}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-9\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{28}\right)-\dfrac{61}{7}+\left(-9\right)\sqrt{49}+\left(\dfrac{73}{2}\right)\sqrt{63}+\left(-\dfrac{3}{2}\right)\sqrt{175}+\left(-\dfrac{17}{7}\right)\sqrt{49}+\left(-\dfrac{17}{7}\right)\sqrt{49}-\dfrac{50}{9}+\left(-2\right)\sqrt{175}+\dfrac{15}{4}+\left(-\dfrac{74}{3}\right)\sqrt{175}-1+\left(\dfrac{36}{7}\right)\sqrt{175}\right)\times\left(\left(\dfrac{19}{4}\right)\sqrt{63}+\left(\left(\dfrac{38}{3}\right)\sqrt{28}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{49}\right)-\left(\left(4\right)\sqrt{175}\right)\right)\\
&=&\left(-63-\left(\left(19\right)\sqrt{7}\right)-\dfrac{61}{7}-63+\left(\dfrac{219}{2}\right)\sqrt{7}+\left(-\dfrac{15}{2}\right)\sqrt{7}-17-17-\dfrac{50}{9}+\left(-10\right)\sqrt{7}+\dfrac{15}{4}+\left(-\dfrac{370}{3}\right)\sqrt{7}-1+\left(\dfrac{180}{7}\right)\sqrt{7}\right)\times\left(\left(\dfrac{57}{4}\right)\sqrt{7}+\left(\left(\dfrac{76}{3}\right)\sqrt{7}\right)+\dfrac{245}{8}-\left(\left(20\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{43223}{252}+\left(-\dfrac{517}{21}\right)\sqrt{7}\right)\left(\left(\dfrac{235}{12}\right)\sqrt{7}+\dfrac{245}{8}\right)\\
&=&\left(-\dfrac{12437375}{3024}\right)\sqrt{7}-\dfrac{1512805}{288}+\left(-\dfrac{121495}{252}\right)\sqrt{49}\\
&=&\left(-\dfrac{12437375}{3024}\right)\sqrt{7}-\dfrac{276085}{32}\\
\end{eqnarray*}