L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(-\dfrac{77}{2}\right)\sqrt{9}+\left(-\dfrac{29}{7}\right)\sqrt{12}+\left(-\dfrac{67}{5}\right)\sqrt{75}+\left(\dfrac{19}{4}\right)\sqrt{9}+\left(-\dfrac{20}{3}\right)\sqrt{12}+1+\dfrac{1}{7}+\left(-\dfrac{64}{9}\right)\sqrt{27}+\left(\dfrac{26}{3}\right)\sqrt{12}+\left(-\dfrac{67}{5}\right)\sqrt{75}+\left(\dfrac{37}{7}\right)\sqrt{12}\) et \( Y=\left(\dfrac{34}{5}\right)\sqrt{75}+\left(-\dfrac{17}{6}\right)\sqrt{12}-3+\left(\dfrac{40}{7}\right)\sqrt{9}+\left(\dfrac{21}{2}\right)\sqrt{9}+\left(-4\right)\sqrt{75}+\dfrac{19}{5}+\left(\dfrac{5}{4}\right)\sqrt{9}+\left(-9\right)\sqrt{75}+\left(\dfrac{34}{5}\right)\sqrt{75}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{77}{2}\right)\sqrt{9}+\left(-\dfrac{29}{7}\right)\sqrt{12}+\left(-\dfrac{67}{5}\right)\sqrt{75}+\left(\dfrac{19}{4}\right)\sqrt{9}+\left(-\dfrac{20}{3}\right)\sqrt{12}+1+\dfrac{1}{7}+\left(-\dfrac{64}{9}\right)\sqrt{27}+\left(\dfrac{26}{3}\right)\sqrt{12}+\left(-\dfrac{67}{5}\right)\sqrt{75}+\left(\dfrac{37}{7}\right)\sqrt{12}\right)+\left(\left(\dfrac{34}{5}\right)\sqrt{75}+\left(-\dfrac{17}{6}\right)\sqrt{12}-3+\left(\dfrac{40}{7}\right)\sqrt{9}+\left(\dfrac{21}{2}\right)\sqrt{9}+\left(-4\right)\sqrt{75}+\dfrac{19}{5}+\left(\dfrac{5}{4}\right)\sqrt{9}+\left(-9\right)\sqrt{75}+\left(\dfrac{34}{5}\right)\sqrt{75}\right)\\
&=&\left(-\dfrac{231}{2}+\left(-\dfrac{58}{7}\right)\sqrt{3}+\left(-67\right)\sqrt{3}+\dfrac{57}{4}+\left(-\dfrac{40}{3}\right)\sqrt{3}+1+\dfrac{1}{7}+\left(-\dfrac{64}{3}\right)\sqrt{3}+\left(\dfrac{52}{3}\right)\sqrt{3}+\left(-67\right)\sqrt{3}+\left(\dfrac{74}{7}\right)\sqrt{3}\right)+\left(\left(34\right)\sqrt{3}+\left(-\dfrac{17}{3}\right)\sqrt{3}-3+\dfrac{120}{7}+\dfrac{63}{2}+\left(-20\right)\sqrt{3}+\dfrac{19}{5}+\dfrac{15}{4}+\left(-45\right)\sqrt{3}+\left(34\right)\sqrt{3}\right)\\
&=&-\dfrac{231}{2}+\left(-\dfrac{58}{7}\right)\sqrt{3}+\left(-67\right)\sqrt{3}+\dfrac{57}{4}+\left(-\dfrac{40}{3}\right)\sqrt{3}+1+\dfrac{1}{7}+\left(-\dfrac{64}{3}\right)\sqrt{3}+\left(\dfrac{52}{3}\right)\sqrt{3}+\left(-67\right)\sqrt{3}+\left(\dfrac{74}{7}\right)\sqrt{3}+\left(34\right)\sqrt{3}+\left(-\dfrac{17}{3}\right)\sqrt{3}-3+\dfrac{120}{7}+\dfrac{63}{2}+\left(-20\right)\sqrt{3}+\dfrac{19}{5}+\dfrac{15}{4}+\left(-45\right)\sqrt{3}+\left(34\right)\sqrt{3}\\
&=&-\dfrac{1642}{35}+\left(-\dfrac{1062}{7}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{77}{2}\right)\sqrt{9}+\left(-\dfrac{29}{7}\right)\sqrt{12}+\left(-\dfrac{67}{5}\right)\sqrt{75}+\left(\dfrac{19}{4}\right)\sqrt{9}+\left(-\dfrac{20}{3}\right)\sqrt{12}+1+\dfrac{1}{7}+\left(-\dfrac{64}{9}\right)\sqrt{27}+\left(\dfrac{26}{3}\right)\sqrt{12}+\left(-\dfrac{67}{5}\right)\sqrt{75}+\left(\dfrac{37}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{75}+\left(-\dfrac{17}{6}\right)\sqrt{12}-3+\left(\dfrac{40}{7}\right)\sqrt{9}+\left(\dfrac{21}{2}\right)\sqrt{9}+\left(-4\right)\sqrt{75}+\dfrac{19}{5}+\left(\dfrac{5}{4}\right)\sqrt{9}+\left(-9\right)\sqrt{75}+\left(\dfrac{34}{5}\right)\sqrt{75}\right)\\
&=&\left(-\dfrac{231}{2}+\left(-\dfrac{58}{7}\right)\sqrt{3}+\left(-67\right)\sqrt{3}+\dfrac{57}{4}+\left(-\dfrac{40}{3}\right)\sqrt{3}+1+\dfrac{1}{7}+\left(-\dfrac{64}{3}\right)\sqrt{3}+\left(\dfrac{52}{3}\right)\sqrt{3}+\left(-67\right)\sqrt{3}+\left(\dfrac{74}{7}\right)\sqrt{3}\right)-\left(\left(34\right)\sqrt{3}+\left(-\dfrac{17}{3}\right)\sqrt{3}-3+\dfrac{120}{7}+\dfrac{63}{2}+\left(-20\right)\sqrt{3}+\dfrac{19}{5}+\dfrac{15}{4}+\left(-45\right)\sqrt{3}+\left(34\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{2803}{28}+\left(-\dfrac{3130}{21}\right)\sqrt{3}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{3}+\dfrac{7447}{140}\right)\\
&=&-\dfrac{2803}{28}+\left(-\dfrac{3130}{21}\right)\sqrt{3}+\left(\dfrac{8}{3}\right)\sqrt{3}-\dfrac{7447}{140}\\
&=&-\dfrac{1533}{10}+\left(-\dfrac{3074}{21}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{77}{2}\right)\sqrt{9}+\left(-\dfrac{29}{7}\right)\sqrt{12}+\left(-\dfrac{67}{5}\right)\sqrt{75}+\left(\dfrac{19}{4}\right)\sqrt{9}+\left(-\dfrac{20}{3}\right)\sqrt{12}+1+\dfrac{1}{7}+\left(-\dfrac{64}{9}\right)\sqrt{27}+\left(\dfrac{26}{3}\right)\sqrt{12}+\left(-\dfrac{67}{5}\right)\sqrt{75}+\left(\dfrac{37}{7}\right)\sqrt{12}\right)\times\left(\left(\dfrac{34}{5}\right)\sqrt{75}+\left(-\dfrac{17}{6}\right)\sqrt{12}-3+\left(\dfrac{40}{7}\right)\sqrt{9}+\left(\dfrac{21}{2}\right)\sqrt{9}+\left(-4\right)\sqrt{75}+\dfrac{19}{5}+\left(\dfrac{5}{4}\right)\sqrt{9}+\left(-9\right)\sqrt{75}+\left(\dfrac{34}{5}\right)\sqrt{75}\right)\\
&=&\left(-\dfrac{231}{2}+\left(-\dfrac{58}{7}\right)\sqrt{3}+\left(-67\right)\sqrt{3}+\dfrac{57}{4}+\left(-\dfrac{40}{3}\right)\sqrt{3}+1+\dfrac{1}{7}+\left(-\dfrac{64}{3}\right)\sqrt{3}+\left(\dfrac{52}{3}\right)\sqrt{3}+\left(-67\right)\sqrt{3}+\left(\dfrac{74}{7}\right)\sqrt{3}\right)\times\left(\left(34\right)\sqrt{3}+\left(-\dfrac{17}{3}\right)\sqrt{3}-3+\dfrac{120}{7}+\dfrac{63}{2}+\left(-20\right)\sqrt{3}+\dfrac{19}{5}+\dfrac{15}{4}+\left(-45\right)\sqrt{3}+\left(34\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{2803}{28}+\left(-\dfrac{3130}{21}\right)\sqrt{3}\right)\left(\left(-\dfrac{8}{3}\right)\sqrt{3}+\dfrac{7447}{140}\right)\\
&=&\left(-\dfrac{750809}{98}\right)\sqrt{3}-\dfrac{20873941}{3920}+\left(\dfrac{25040}{63}\right)\sqrt{9}\\
&=&\left(-\dfrac{750809}{98}\right)\sqrt{3}-\dfrac{48599423}{11760}\\
\end{eqnarray*}