L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{66}{7}\right)\sqrt{75}+\left(\dfrac{7}{2}\right)\sqrt{9}+\left(\dfrac{65}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{47}{7}\right)\sqrt{12}+\dfrac{38}{7}+\left(-3\right)\sqrt{12}+\dfrac{17}{8}+\left(\dfrac{73}{5}\right)\sqrt{75}\right)\) et \( Y=\left(-4\right)\sqrt{27}\) . 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{66}{7}\right)\sqrt{75}+\left(\dfrac{7}{2}\right)\sqrt{9}+\left(\dfrac{65}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{47}{7}\right)\sqrt{12}+\dfrac{38}{7}+\left(-3\right)\sqrt{12}+\dfrac{17}{8}+\left(\dfrac{73}{5}\right)\sqrt{75}\right)\right)+\left(\left(-4\right)\sqrt{27}\right)\\
&=&\left(\left(\left(-\dfrac{330}{7}\right)\sqrt{3}+\dfrac{21}{2}+\left(\dfrac{195}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{94}{7}\right)\sqrt{3}+\dfrac{38}{7}+\left(-6\right)\sqrt{3}+\dfrac{17}{8}+\left(73\right)\sqrt{3}\right)\right)+\left(\left(-12\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{330}{7}\right)\sqrt{3}+\dfrac{21}{2}+\left(\dfrac{195}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{94}{7}\right)\sqrt{3}+\dfrac{38}{7}+\left(-6\right)\sqrt{3}+\dfrac{17}{8}+\left(73\right)\sqrt{3}\right)+\left(-12\right)\sqrt{3}\\
&=&\left(-\dfrac{213}{14}\right)\sqrt{3}+\dfrac{165}{56}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{66}{7}\right)\sqrt{75}+\left(\dfrac{7}{2}\right)\sqrt{9}+\left(\dfrac{65}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{47}{7}\right)\sqrt{12}+\dfrac{38}{7}+\left(-3\right)\sqrt{12}+\dfrac{17}{8}+\left(\dfrac{73}{5}\right)\sqrt{75}\right)\right)-\left(\left(-4\right)\sqrt{27}\right)\\
&=&\left(\left(\left(-\dfrac{330}{7}\right)\sqrt{3}+\dfrac{21}{2}+\left(\dfrac{195}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{94}{7}\right)\sqrt{3}+\dfrac{38}{7}+\left(-6\right)\sqrt{3}+\dfrac{17}{8}+\left(73\right)\sqrt{3}\right)\right)-\left(\left(-12\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{45}{14}\right)\sqrt{3}+\dfrac{165}{56}\right)-\left(\left(-12\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{45}{14}\right)\sqrt{3}+\dfrac{165}{56}+\left(12\right)\sqrt{3}\\
&=&\left(\dfrac{123}{14}\right)\sqrt{3}+\dfrac{165}{56}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{66}{7}\right)\sqrt{75}+\left(\dfrac{7}{2}\right)\sqrt{9}+\left(\dfrac{65}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{47}{7}\right)\sqrt{12}+\dfrac{38}{7}+\left(-3\right)\sqrt{12}+\dfrac{17}{8}+\left(\dfrac{73}{5}\right)\sqrt{75}\right)\right)\times\left(\left(-4\right)\sqrt{27}\right)\\
&=&\left(\left(\left(-\dfrac{330}{7}\right)\sqrt{3}+\dfrac{21}{2}+\left(\dfrac{195}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{94}{7}\right)\sqrt{3}+\dfrac{38}{7}+\left(-6\right)\sqrt{3}+\dfrac{17}{8}+\left(73\right)\sqrt{3}\right)\right)\times\left(\left(-12\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{45}{14}\right)\sqrt{3}+\dfrac{165}{56}\right)\left(\left(-12\right)\sqrt{3}\right)\\
&=&\left(\dfrac{270}{7}\right)\sqrt{9}+\left(-\dfrac{495}{14}\right)\sqrt{3}\\
&=&\dfrac{810}{7}+\left(-\dfrac{495}{14}\right)\sqrt{3}\\
\end{eqnarray*}