L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(6\right)\sqrt{75}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{12}\right)\) et \( Y=\left(\left(\dfrac{37}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{12}\right)+\left(-\dfrac{24}{5}\right)\sqrt{27}+\left(\dfrac{1}{8}\right)\sqrt{27}+\left(\left(\dfrac{21}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{27}\right)-\left(\left(-9\right)\sqrt{9}\right)\) . 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(6\right)\sqrt{75}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{12}\right)\right)+\left(\left(\left(\dfrac{37}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{12}\right)+\left(-\dfrac{24}{5}\right)\sqrt{27}+\left(\dfrac{1}{8}\right)\sqrt{27}+\left(\left(\dfrac{21}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{27}\right)-\left(\left(-9\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(30\right)\sqrt{3}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}\right)\right)+\left(\left(\left(37\right)\sqrt{3}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{3}\right)+\left(-\dfrac{72}{5}\right)\sqrt{3}+\left(\dfrac{3}{8}\right)\sqrt{3}+\left(\left(21\right)\sqrt{3}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{3}\right)+27\right)\\
&=&\left(\left(30\right)\sqrt{3}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}\right)+\left(\left(37\right)\sqrt{3}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{3}\right)+\left(-\dfrac{72}{5}\right)\sqrt{3}+\left(\dfrac{3}{8}\right)\sqrt{3}+\left(\left(21\right)\sqrt{3}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{3}\right)+27\\
&=&\left(\dfrac{2619}{40}\right)\sqrt{3}+27\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(6\right)\sqrt{75}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{12}\right)\right)-\left(\left(\left(\dfrac{37}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{12}\right)+\left(-\dfrac{24}{5}\right)\sqrt{27}+\left(\dfrac{1}{8}\right)\sqrt{27}+\left(\left(\dfrac{21}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{27}\right)-\left(\left(-9\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(30\right)\sqrt{3}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}\right)\right)-\left(\left(\left(37\right)\sqrt{3}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{3}\right)+\left(-\dfrac{72}{5}\right)\sqrt{3}+\left(\dfrac{3}{8}\right)\sqrt{3}+\left(\left(21\right)\sqrt{3}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{3}\right)+27\right)\\
&=&\left(\left(\dfrac{134}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{2497}{120}\right)\sqrt{3}+27\right)\\
&=&\left(\dfrac{134}{3}\right)\sqrt{3}+\left(-\dfrac{2497}{120}\right)\sqrt{3}-27\\
&=&\left(\dfrac{2863}{120}\right)\sqrt{3}-27\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(6\right)\sqrt{75}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{12}\right)\right)\times\left(\left(\left(\dfrac{37}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{12}\right)+\left(-\dfrac{24}{5}\right)\sqrt{27}+\left(\dfrac{1}{8}\right)\sqrt{27}+\left(\left(\dfrac{21}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{27}\right)-\left(\left(-9\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(30\right)\sqrt{3}\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}\right)\right)\times\left(\left(\left(37\right)\sqrt{3}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{3}\right)+\left(-\dfrac{72}{5}\right)\sqrt{3}+\left(\dfrac{3}{8}\right)\sqrt{3}+\left(\left(21\right)\sqrt{3}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{3}\right)+27\right)\\
&=&\left(\left(\dfrac{134}{3}\right)\sqrt{3}\right)\left(\left(\dfrac{2497}{120}\right)\sqrt{3}+27\right)\\
&=&\left(\dfrac{167299}{180}\right)\sqrt{9}+\left(1206\right)\sqrt{3}\\
&=&\dfrac{167299}{60}+\left(1206\right)\sqrt{3}\\
\end{eqnarray*}