L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{14}{3}\right)\sqrt{27}\) et \( Y=\left(\left(-\dfrac{71}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{5}\right)\sqrt{9}+\left(-\dfrac{33}{2}\right)\sqrt{75}+\left(\dfrac{7}{3}\right)\sqrt{75}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{14}{3}\right)\sqrt{27}\right)+\left(\left(\left(-\dfrac{71}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{5}\right)\sqrt{9}+\left(-\dfrac{33}{2}\right)\sqrt{75}+\left(\dfrac{7}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-14\right)\sqrt{3}\right)+\left(\left(\left(-71\right)\sqrt{3}\right)-\left(-\dfrac{3}{5}+\left(-\dfrac{165}{2}\right)\sqrt{3}+\left(\dfrac{35}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(-14\right)\sqrt{3}+\left(\left(-71\right)\sqrt{3}\right)-\left(-\dfrac{3}{5}+\left(-\dfrac{165}{2}\right)\sqrt{3}+\left(\dfrac{35}{3}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{85}{6}\right)\sqrt{3}+\dfrac{3}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{14}{3}\right)\sqrt{27}\right)-\left(\left(\left(-\dfrac{71}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{5}\right)\sqrt{9}+\left(-\dfrac{33}{2}\right)\sqrt{75}+\left(\dfrac{7}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-14\right)\sqrt{3}\right)-\left(\left(\left(-71\right)\sqrt{3}\right)-\left(-\dfrac{3}{5}+\left(-\dfrac{165}{2}\right)\sqrt{3}+\left(\dfrac{35}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-14\right)\sqrt{3}\right)-\left(\left(-\dfrac{1}{6}\right)\sqrt{3}+\dfrac{3}{5}\right)\\
&=&\left(-14\right)\sqrt{3}+\left(\dfrac{1}{6}\right)\sqrt{3}-\dfrac{3}{5}\\
&=&\left(-\dfrac{83}{6}\right)\sqrt{3}-\dfrac{3}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{14}{3}\right)\sqrt{27}\right)\times\left(\left(\left(-\dfrac{71}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{5}\right)\sqrt{9}+\left(-\dfrac{33}{2}\right)\sqrt{75}+\left(\dfrac{7}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-14\right)\sqrt{3}\right)\times\left(\left(\left(-71\right)\sqrt{3}\right)-\left(-\dfrac{3}{5}+\left(-\dfrac{165}{2}\right)\sqrt{3}+\left(\dfrac{35}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-14\right)\sqrt{3}\right)\left(\left(-\dfrac{1}{6}\right)\sqrt{3}+\dfrac{3}{5}\right)\\
&=&\left(\dfrac{7}{3}\right)\sqrt{9}+\left(-\dfrac{42}{5}\right)\sqrt{3}\\
&=&7+\left(-\dfrac{42}{5}\right)\sqrt{3}\\
\end{eqnarray*}