L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(-1\right)\sqrt{27}+\left(-\dfrac{39}{8}\right)\sqrt{12}+\left(-\dfrac{23}{2}\right)\sqrt{27}+\left(\left(-\dfrac{40}{7}\right)\sqrt{75}\right)-\left(\left(3\right)\sqrt{75}\right)-\dfrac{27}{4}+\left(-9\right)\sqrt{27}+\left(7\right)\sqrt{9}\) et \( Y=\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{32}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{52}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{9}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-1\right)\sqrt{27}+\left(-\dfrac{39}{8}\right)\sqrt{12}+\left(-\dfrac{23}{2}\right)\sqrt{27}+\left(\left(-\dfrac{40}{7}\right)\sqrt{75}\right)-\left(\left(3\right)\sqrt{75}\right)-\dfrac{27}{4}+\left(-9\right)\sqrt{27}+\left(7\right)\sqrt{9}\right)+\left(\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{32}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{52}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(-3\right)\sqrt{3}+\left(-\dfrac{39}{4}\right)\sqrt{3}+\left(-\dfrac{69}{2}\right)\sqrt{3}+\left(\left(-\dfrac{200}{7}\right)\sqrt{3}\right)-\left(\left(15\right)\sqrt{3}\right)-\dfrac{27}{4}+\left(-27\right)\sqrt{3}+21\right)+\left(\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{160}{3}\right)\sqrt{3}\right)-\dfrac{156}{5}+\dfrac{153}{5}\right)\right)\\
&=&\left(-3\right)\sqrt{3}+\left(-\dfrac{39}{4}\right)\sqrt{3}+\left(-\dfrac{69}{2}\right)\sqrt{3}+\left(\left(-\dfrac{200}{7}\right)\sqrt{3}\right)-\left(\left(15\right)\sqrt{3}\right)-\dfrac{27}{4}+\left(-27\right)\sqrt{3}+21+\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{160}{3}\right)\sqrt{3}\right)-\dfrac{156}{5}+\dfrac{153}{5}\right)\\
&=&\left(-\dfrac{13817}{84}\right)\sqrt{3}+\dfrac{297}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-1\right)\sqrt{27}+\left(-\dfrac{39}{8}\right)\sqrt{12}+\left(-\dfrac{23}{2}\right)\sqrt{27}+\left(\left(-\dfrac{40}{7}\right)\sqrt{75}\right)-\left(\left(3\right)\sqrt{75}\right)-\dfrac{27}{4}+\left(-9\right)\sqrt{27}+\left(7\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{32}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{52}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(-3\right)\sqrt{3}+\left(-\dfrac{39}{4}\right)\sqrt{3}+\left(-\dfrac{69}{2}\right)\sqrt{3}+\left(\left(-\dfrac{200}{7}\right)\sqrt{3}\right)-\left(\left(15\right)\sqrt{3}\right)-\dfrac{27}{4}+\left(-27\right)\sqrt{3}+21\right)-\left(\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{160}{3}\right)\sqrt{3}\right)-\dfrac{156}{5}+\dfrac{153}{5}\right)\right)\\
&=&\left(\left(-\dfrac{3299}{28}\right)\sqrt{3}+\dfrac{57}{4}\right)-\left(\left(-\dfrac{140}{3}\right)\sqrt{3}+\dfrac{3}{5}\right)\\
&=&\left(-\dfrac{3299}{28}\right)\sqrt{3}+\dfrac{57}{4}+\left(\dfrac{140}{3}\right)\sqrt{3}-\dfrac{3}{5}\\
&=&\left(-\dfrac{5977}{84}\right)\sqrt{3}+\dfrac{273}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-1\right)\sqrt{27}+\left(-\dfrac{39}{8}\right)\sqrt{12}+\left(-\dfrac{23}{2}\right)\sqrt{27}+\left(\left(-\dfrac{40}{7}\right)\sqrt{75}\right)-\left(\left(3\right)\sqrt{75}\right)-\dfrac{27}{4}+\left(-9\right)\sqrt{27}+\left(7\right)\sqrt{9}\right)\times\left(\left(\left(\dfrac{10}{3}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{32}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{52}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{51}{5}\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(-3\right)\sqrt{3}+\left(-\dfrac{39}{4}\right)\sqrt{3}+\left(-\dfrac{69}{2}\right)\sqrt{3}+\left(\left(-\dfrac{200}{7}\right)\sqrt{3}\right)-\left(\left(15\right)\sqrt{3}\right)-\dfrac{27}{4}+\left(-27\right)\sqrt{3}+21\right)\times\left(\left(\left(\dfrac{20}{3}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{160}{3}\right)\sqrt{3}\right)-\dfrac{156}{5}+\dfrac{153}{5}\right)\right)\\
&=&\left(\left(-\dfrac{3299}{28}\right)\sqrt{3}+\dfrac{57}{4}\right)\left(\left(-\dfrac{140}{3}\right)\sqrt{3}+\dfrac{3}{5}\right)\\
&=&\left(\dfrac{16495}{3}\right)\sqrt{9}+\left(-\dfrac{102997}{140}\right)\sqrt{3}+\dfrac{171}{20}\\
&=&\dfrac{330071}{20}+\left(-\dfrac{102997}{140}\right)\sqrt{3}\\
\end{eqnarray*}