L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-1-\left(\left(-\dfrac{64}{5}\right)\sqrt{75}\right)-\left(\left(2\right)\sqrt{27}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{75}\right)+\left(2\right)\sqrt{27}+\left(-\dfrac{23}{6}\right)\sqrt{75}\) et \( Y=\left(\left(-\dfrac{53}{7}\right)\sqrt{27}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{27}+\left(\dfrac{25}{3}\right)\sqrt{75}+\left(0\right)\sqrt{9}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-1-\left(\left(-\dfrac{64}{5}\right)\sqrt{75}\right)-\left(\left(2\right)\sqrt{27}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{75}\right)+\left(2\right)\sqrt{27}+\left(-\dfrac{23}{6}\right)\sqrt{75}\right)+\left(\left(\left(-\dfrac{53}{7}\right)\sqrt{27}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{27}+\left(\dfrac{25}{3}\right)\sqrt{75}+\left(0\right)\sqrt{9}\right)\right)\\
&=&\left(-1-\left(\left(-64\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)-\left(\left(-\dfrac{110}{3}\right)\sqrt{3}\right)+\left(6\right)\sqrt{3}+\left(-\dfrac{115}{6}\right)\sqrt{3}\right)+\left(\left(\left(-\dfrac{159}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{141}{8}\right)\sqrt{3}+\left(\dfrac{125}{3}\right)\sqrt{3}+0\right)\right)\\
&=&-1-\left(\left(-64\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)-\left(\left(-\dfrac{110}{3}\right)\sqrt{3}\right)+\left(6\right)\sqrt{3}+\left(-\dfrac{115}{6}\right)\sqrt{3}+\left(\left(-\dfrac{159}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{141}{8}\right)\sqrt{3}+\left(\dfrac{125}{3}\right)\sqrt{3}+0\right)\\
&=&-1+\left(\dfrac{5837}{168}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-1-\left(\left(-\dfrac{64}{5}\right)\sqrt{75}\right)-\left(\left(2\right)\sqrt{27}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{75}\right)+\left(2\right)\sqrt{27}+\left(-\dfrac{23}{6}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{53}{7}\right)\sqrt{27}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{27}+\left(\dfrac{25}{3}\right)\sqrt{75}+\left(0\right)\sqrt{9}\right)\right)\\
&=&\left(-1-\left(\left(-64\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)-\left(\left(-\dfrac{110}{3}\right)\sqrt{3}\right)+\left(6\right)\sqrt{3}+\left(-\dfrac{115}{6}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{159}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{141}{8}\right)\sqrt{3}+\left(\dfrac{125}{3}\right)\sqrt{3}+0\right)\right)\\
&=&\left(-1+\left(\dfrac{163}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{7855}{168}\right)\sqrt{3}+0\right)\\
&=&-1+\left(\dfrac{163}{2}\right)\sqrt{3}+\left(\dfrac{7855}{168}\right)\sqrt{3}+0\\
&=&-1+\left(\dfrac{21547}{168}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-1-\left(\left(-\dfrac{64}{5}\right)\sqrt{75}\right)-\left(\left(2\right)\sqrt{27}\right)-\left(\left(-\dfrac{22}{3}\right)\sqrt{75}\right)+\left(2\right)\sqrt{27}+\left(-\dfrac{23}{6}\right)\sqrt{75}\right)\times\left(\left(\left(-\dfrac{53}{7}\right)\sqrt{27}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{27}+\left(\dfrac{25}{3}\right)\sqrt{75}+\left(0\right)\sqrt{9}\right)\right)\\
&=&\left(-1-\left(\left(-64\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)-\left(\left(-\dfrac{110}{3}\right)\sqrt{3}\right)+\left(6\right)\sqrt{3}+\left(-\dfrac{115}{6}\right)\sqrt{3}\right)\times\left(\left(\left(-\dfrac{159}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{141}{8}\right)\sqrt{3}+\left(\dfrac{125}{3}\right)\sqrt{3}+0\right)\right)\\
&=&\left(-1+\left(\dfrac{163}{2}\right)\sqrt{3}\right)\left(\left(-\dfrac{7855}{168}\right)\sqrt{3}+0\right)\\
&=&\left(\dfrac{7855}{168}\right)\sqrt{3}+0+\left(-\dfrac{1280365}{336}\right)\sqrt{9}\\
&=&\left(\dfrac{7855}{168}\right)\sqrt{3}-\dfrac{1280365}{112}\\
\end{eqnarray*}