L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{9}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)+\dfrac{13}{2}-\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{27}+\left(-\dfrac{7}{2}\right)\sqrt{9}\right)-\left(5-\left(\left(-\dfrac{13}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{41}{9}\right)\sqrt{9}\right)\right)-\left(\left(\left(\dfrac{63}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{9}\right)\right)\) et \( Y=\left(9\right)\sqrt{9}\) . 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{9}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)+\dfrac{13}{2}-\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{27}+\left(-\dfrac{7}{2}\right)\sqrt{9}\right)-\left(5-\left(\left(-\dfrac{13}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{41}{9}\right)\sqrt{9}\right)\right)-\left(\left(\left(\dfrac{63}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{9}\right)\right)\right)+\left(\left(9\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{27}{2}-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)+\dfrac{13}{2}-\left(\left(-43\right)\sqrt{3}\right)\right)-\left(\left(-37\right)\sqrt{3}-\dfrac{21}{2}\right)-\left(5-\left(\left(-\dfrac{39}{7}\right)\sqrt{3}\right)-\dfrac{41}{3}\right)-\left(\dfrac{189}{4}-\left(\left(-5\right)\sqrt{3}\right)+\dfrac{108}{7}\right)\right)+\left(27\right)\\
&=&\left(\dfrac{27}{2}-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)+\dfrac{13}{2}-\left(\left(-43\right)\sqrt{3}\right)\right)-\left(\left(-37\right)\sqrt{3}-\dfrac{21}{2}\right)-\left(5-\left(\left(-\dfrac{39}{7}\right)\sqrt{3}\right)-\dfrac{41}{3}\right)-\left(\dfrac{189}{4}-\left(\left(-5\right)\sqrt{3}\right)+\dfrac{108}{7}\right)+27\\
&=&\dfrac{293}{84}+\left(\dfrac{1318}{21}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{9}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)+\dfrac{13}{2}-\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{27}+\left(-\dfrac{7}{2}\right)\sqrt{9}\right)-\left(5-\left(\left(-\dfrac{13}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{41}{9}\right)\sqrt{9}\right)\right)-\left(\left(\left(\dfrac{63}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{9}\right)\right)\right)-\left(\left(9\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{27}{2}-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)+\dfrac{13}{2}-\left(\left(-43\right)\sqrt{3}\right)\right)-\left(\left(-37\right)\sqrt{3}-\dfrac{21}{2}\right)-\left(5-\left(\left(-\dfrac{39}{7}\right)\sqrt{3}\right)-\dfrac{41}{3}\right)-\left(\dfrac{189}{4}-\left(\left(-5\right)\sqrt{3}\right)+\dfrac{108}{7}\right)\right)-\left(27\right)\\
&=&\left(-\dfrac{1975}{84}+\left(\dfrac{1318}{21}\right)\sqrt{3}\right)-\left(27\right)\\
&=&-\dfrac{1975}{84}+\left(\dfrac{1318}{21}\right)\sqrt{3}+-27\\
&=&-\dfrac{4243}{84}+\left(\dfrac{1318}{21}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{9}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)+\dfrac{13}{2}-\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{27}+\left(-\dfrac{7}{2}\right)\sqrt{9}\right)-\left(5-\left(\left(-\dfrac{13}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{41}{9}\right)\sqrt{9}\right)\right)-\left(\left(\left(\dfrac{63}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{9}\right)\right)\right)\times\left(\left(9\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{27}{2}-\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)+\dfrac{13}{2}-\left(\left(-43\right)\sqrt{3}\right)\right)-\left(\left(-37\right)\sqrt{3}-\dfrac{21}{2}\right)-\left(5-\left(\left(-\dfrac{39}{7}\right)\sqrt{3}\right)-\dfrac{41}{3}\right)-\left(\dfrac{189}{4}-\left(\left(-5\right)\sqrt{3}\right)+\dfrac{108}{7}\right)\right)\times\left(27\right)\\
&=&\left(-\dfrac{1975}{84}+\left(\dfrac{1318}{21}\right)\sqrt{3}\right)\left(27\right)\\
&=&-\dfrac{17775}{28}+\left(\dfrac{11862}{7}\right)\sqrt{3}\\
&=&-\dfrac{17775}{28}+\left(\dfrac{11862}{7}\right)\sqrt{3}\\
\end{eqnarray*}