L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{9}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{75}\right)+\left(-\dfrac{28}{3}\right)\sqrt{75}\) et \( Y=\left(\left(-\dfrac{55}{3}\right)\sqrt{27}+\left(9\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{20}{7}\right)\sqrt{27}\right)-\left(\left(-3\right)\sqrt{12}\right)-\left(\left(\dfrac{53}{7}\right)\sqrt{75}\right)-\left(\left(0\right)\sqrt{27}\right)\right)\) . 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(\dfrac{9}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{75}\right)+\left(-\dfrac{28}{3}\right)\sqrt{75}\right)+\left(\left(\left(-\dfrac{55}{3}\right)\sqrt{27}+\left(9\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{20}{7}\right)\sqrt{27}\right)-\left(\left(-3\right)\sqrt{12}\right)-\left(\left(\dfrac{53}{7}\right)\sqrt{75}\right)-\left(\left(0\right)\sqrt{27}\right)\right)\right)\\
&=&\left(\dfrac{27}{4}-\left(\left(-\dfrac{75}{2}\right)\sqrt{3}\right)+\left(-\dfrac{140}{3}\right)\sqrt{3}\right)+\left(\left(\left(-55\right)\sqrt{3}+27\right)-\left(\left(\left(\dfrac{60}{7}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)-\left(\left(\dfrac{265}{7}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)\right)\\
&=&\dfrac{27}{4}-\left(\left(-\dfrac{75}{2}\right)\sqrt{3}\right)+\left(-\dfrac{140}{3}\right)\sqrt{3}+\left(\left(-55\right)\sqrt{3}+27\right)-\left(\left(\left(\dfrac{60}{7}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)-\left(\left(\dfrac{265}{7}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)\\
&=&\dfrac{135}{4}+\left(-\dfrac{1717}{42}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{9}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{75}\right)+\left(-\dfrac{28}{3}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{55}{3}\right)\sqrt{27}+\left(9\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{20}{7}\right)\sqrt{27}\right)-\left(\left(-3\right)\sqrt{12}\right)-\left(\left(\dfrac{53}{7}\right)\sqrt{75}\right)-\left(\left(0\right)\sqrt{27}\right)\right)\right)\\
&=&\left(\dfrac{27}{4}-\left(\left(-\dfrac{75}{2}\right)\sqrt{3}\right)+\left(-\dfrac{140}{3}\right)\sqrt{3}\right)-\left(\left(\left(-55\right)\sqrt{3}+27\right)-\left(\left(\left(\dfrac{60}{7}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)-\left(\left(\dfrac{265}{7}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\dfrac{27}{4}+\left(-\dfrac{55}{6}\right)\sqrt{3}\right)-\left(\left(-\dfrac{222}{7}\right)\sqrt{3}+27\right)\\
&=&\dfrac{27}{4}+\left(-\dfrac{55}{6}\right)\sqrt{3}+\left(\dfrac{222}{7}\right)\sqrt{3}-27\\
&=&-\dfrac{81}{4}+\left(\dfrac{947}{42}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{9}{4}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{75}\right)+\left(-\dfrac{28}{3}\right)\sqrt{75}\right)\times\left(\left(\left(-\dfrac{55}{3}\right)\sqrt{27}+\left(9\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{20}{7}\right)\sqrt{27}\right)-\left(\left(-3\right)\sqrt{12}\right)-\left(\left(\dfrac{53}{7}\right)\sqrt{75}\right)-\left(\left(0\right)\sqrt{27}\right)\right)\right)\\
&=&\left(\dfrac{27}{4}-\left(\left(-\dfrac{75}{2}\right)\sqrt{3}\right)+\left(-\dfrac{140}{3}\right)\sqrt{3}\right)\times\left(\left(\left(-55\right)\sqrt{3}+27\right)-\left(\left(\left(\dfrac{60}{7}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)-\left(\left(\dfrac{265}{7}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\dfrac{27}{4}+\left(-\dfrac{55}{6}\right)\sqrt{3}\right)\left(\left(-\dfrac{222}{7}\right)\sqrt{3}+27\right)\\
&=&\left(-\dfrac{3231}{7}\right)\sqrt{3}+\dfrac{729}{4}+\left(\dfrac{2035}{7}\right)\sqrt{9}\\
&=&\left(-\dfrac{3231}{7}\right)\sqrt{3}+\dfrac{29523}{28}\\
\end{eqnarray*}