L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=8-\left(\left(-2\right)\sqrt{9}\right)\) et \( Y=\left(\left(\dfrac{61}{5}\right)\sqrt{27}\right)-\left(\left(\left(-\dfrac{5}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{9}\right)-\dfrac{61}{2}-\left(\left(-\dfrac{11}{4}\right)\sqrt{27}\right)\right)-\left(\left(\left(1\right)\sqrt{75}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{27}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(8-\left(\left(-2\right)\sqrt{9}\right)\right)+\left(\left(\left(\dfrac{61}{5}\right)\sqrt{27}\right)-\left(\left(\left(-\dfrac{5}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{9}\right)-\dfrac{61}{2}-\left(\left(-\dfrac{11}{4}\right)\sqrt{27}\right)\right)-\left(\left(\left(1\right)\sqrt{75}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{27}\right)\right)\right)\\
&=&\left(8+6\right)+\left(\left(\left(\dfrac{183}{5}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{10}{7}\right)\sqrt{3}\right)-5-\dfrac{61}{2}-\left(\left(-\dfrac{33}{4}\right)\sqrt{3}\right)\right)-\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-\dfrac{136}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{3}\right)\right)\right)\\
&=&8+6+\left(\left(\dfrac{183}{5}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{10}{7}\right)\sqrt{3}\right)-5-\dfrac{61}{2}-\left(\left(-\dfrac{33}{4}\right)\sqrt{3}\right)\right)-\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-\dfrac{136}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{3}\right)\right)\\
&=&\dfrac{99}{2}+\left(-\dfrac{35069}{1260}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(8-\left(\left(-2\right)\sqrt{9}\right)\right)-\left(\left(\left(\dfrac{61}{5}\right)\sqrt{27}\right)-\left(\left(\left(-\dfrac{5}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{9}\right)-\dfrac{61}{2}-\left(\left(-\dfrac{11}{4}\right)\sqrt{27}\right)\right)-\left(\left(\left(1\right)\sqrt{75}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{27}\right)\right)\right)\\
&=&\left(8+6\right)-\left(\left(\left(\dfrac{183}{5}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{10}{7}\right)\sqrt{3}\right)-5-\dfrac{61}{2}-\left(\left(-\dfrac{33}{4}\right)\sqrt{3}\right)\right)-\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-\dfrac{136}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(14\right)-\left(\left(-\dfrac{35069}{1260}\right)\sqrt{3}+\dfrac{71}{2}\right)\\
&=&14+\left(\dfrac{35069}{1260}\right)\sqrt{3}-\dfrac{71}{2}\\
&=&-\dfrac{43}{2}+\left(\dfrac{35069}{1260}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(8-\left(\left(-2\right)\sqrt{9}\right)\right)\times\left(\left(\left(\dfrac{61}{5}\right)\sqrt{27}\right)-\left(\left(\left(-\dfrac{5}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{9}\right)-\dfrac{61}{2}-\left(\left(-\dfrac{11}{4}\right)\sqrt{27}\right)\right)-\left(\left(\left(1\right)\sqrt{75}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{27}\right)\right)\right)\\
&=&\left(8+6\right)\times\left(\left(\left(\dfrac{183}{5}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{10}{7}\right)\sqrt{3}\right)-5-\dfrac{61}{2}-\left(\left(-\dfrac{33}{4}\right)\sqrt{3}\right)\right)-\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-\dfrac{136}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(14\right)\left(\left(-\dfrac{35069}{1260}\right)\sqrt{3}+\dfrac{71}{2}\right)\\
&=&\left(-\dfrac{35069}{90}\right)\sqrt{3}+497\\
&=&\left(-\dfrac{35069}{90}\right)\sqrt{3}+497\\
\end{eqnarray*}