L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\left(-\dfrac{61}{4}\right)\sqrt{175}\right)-\left(\left(-\dfrac{50}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{67}{8}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{49}+\left(-5\right)\sqrt{28}\right)\) et \( Y=\left(\left(\left(-2\right)\sqrt{175}\right)-\left(\left(1\right)\sqrt{63}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{63}\right)-\dfrac{4}{3}-\left(\left(-\dfrac{32}{3}\right)\sqrt{49}\right)\right)-\left(\left(-2\right)\sqrt{28}\right)-\left(\left(\dfrac{51}{8}\right)\sqrt{175}\right)\) . 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(\left(-\dfrac{61}{4}\right)\sqrt{175}\right)-\left(\left(-\dfrac{50}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{67}{8}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{49}+\left(-5\right)\sqrt{28}\right)\right)+\left(\left(\left(\left(-2\right)\sqrt{175}\right)-\left(\left(1\right)\sqrt{63}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{63}\right)-\dfrac{4}{3}-\left(\left(-\dfrac{32}{3}\right)\sqrt{49}\right)\right)-\left(\left(-2\right)\sqrt{28}\right)-\left(\left(\dfrac{51}{8}\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{305}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{100}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{201}{8}\right)\sqrt{7}\right)-\left(\left(2\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{93}{2}\right)\sqrt{7}\right)-\left(-\dfrac{49}{4}+\left(-10\right)\sqrt{7}\right)\right)+\left(\left(\left(\left(-10\right)\sqrt{7}\right)-\left(\left(3\right)\sqrt{7}\right)-\left(\left(-5\right)\sqrt{7}\right)-\dfrac{4}{3}+\dfrac{224}{3}\right)-\left(\left(-4\right)\sqrt{7}\right)-\left(\left(\dfrac{255}{8}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\left(-\dfrac{305}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{100}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{201}{8}\right)\sqrt{7}\right)-\left(\left(2\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{93}{2}\right)\sqrt{7}\right)-\left(-\dfrac{49}{4}+\left(-10\right)\sqrt{7}\right)+\left(\left(\left(-10\right)\sqrt{7}\right)-\left(\left(3\right)\sqrt{7}\right)-\left(\left(-5\right)\sqrt{7}\right)-\dfrac{4}{3}+\dfrac{224}{3}\right)-\left(\left(-4\right)\sqrt{7}\right)-\left(\left(\dfrac{255}{8}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{1917}{28}\right)\sqrt{7}+\dfrac{1027}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{61}{4}\right)\sqrt{175}\right)-\left(\left(-\dfrac{50}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{67}{8}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{49}+\left(-5\right)\sqrt{28}\right)\right)-\left(\left(\left(\left(-2\right)\sqrt{175}\right)-\left(\left(1\right)\sqrt{63}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{63}\right)-\dfrac{4}{3}-\left(\left(-\dfrac{32}{3}\right)\sqrt{49}\right)\right)-\left(\left(-2\right)\sqrt{28}\right)-\left(\left(\dfrac{51}{8}\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{305}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{100}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{201}{8}\right)\sqrt{7}\right)-\left(\left(2\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{93}{2}\right)\sqrt{7}\right)-\left(-\dfrac{49}{4}+\left(-10\right)\sqrt{7}\right)\right)-\left(\left(\left(\left(-10\right)\sqrt{7}\right)-\left(\left(3\right)\sqrt{7}\right)-\left(\left(-5\right)\sqrt{7}\right)-\dfrac{4}{3}+\dfrac{224}{3}\right)-\left(\left(-4\right)\sqrt{7}\right)-\left(\left(\dfrac{255}{8}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{1825}{56}\right)\sqrt{7}+\dfrac{49}{4}\right)-\left(\left(-\dfrac{287}{8}\right)\sqrt{7}+\dfrac{220}{3}\right)\\
&=&\left(-\dfrac{1825}{56}\right)\sqrt{7}+\dfrac{49}{4}+\left(\dfrac{287}{8}\right)\sqrt{7}-\dfrac{220}{3}\\
&=&\left(\dfrac{23}{7}\right)\sqrt{7}-\dfrac{733}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{61}{4}\right)\sqrt{175}\right)-\left(\left(-\dfrac{50}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{67}{8}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{49}+\left(-5\right)\sqrt{28}\right)\right)\times\left(\left(\left(\left(-2\right)\sqrt{175}\right)-\left(\left(1\right)\sqrt{63}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{63}\right)-\dfrac{4}{3}-\left(\left(-\dfrac{32}{3}\right)\sqrt{49}\right)\right)-\left(\left(-2\right)\sqrt{28}\right)-\left(\left(\dfrac{51}{8}\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{305}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{100}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{201}{8}\right)\sqrt{7}\right)-\left(\left(2\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{93}{2}\right)\sqrt{7}\right)-\left(-\dfrac{49}{4}+\left(-10\right)\sqrt{7}\right)\right)\times\left(\left(\left(\left(-10\right)\sqrt{7}\right)-\left(\left(3\right)\sqrt{7}\right)-\left(\left(-5\right)\sqrt{7}\right)-\dfrac{4}{3}+\dfrac{224}{3}\right)-\left(\left(-4\right)\sqrt{7}\right)-\left(\left(\dfrac{255}{8}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{1825}{56}\right)\sqrt{7}+\dfrac{49}{4}\right)\left(\left(-\dfrac{287}{8}\right)\sqrt{7}+\dfrac{220}{3}\right)\\
&=&\left(\dfrac{74825}{64}\right)\sqrt{49}+\left(-\dfrac{1901323}{672}\right)\sqrt{7}+\dfrac{2695}{3}\\
&=&\dfrac{1743805}{192}+\left(-\dfrac{1901323}{672}\right)\sqrt{7}\\
\end{eqnarray*}