L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{71}{8}\right)\sqrt{27}\) et \( Y=\left(\left(\dfrac{54}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{4}\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{75}\right)+\left(\left(-\dfrac{41}{2}\right)\sqrt{9}\right)+2-\left(\left(\dfrac{59}{4}\right)\sqrt{75}\right)+\left(-\dfrac{33}{5}\right)\sqrt{75}+\left(\dfrac{64}{3}\right)\sqrt{9}\) . 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{71}{8}\right)\sqrt{27}\right)+\left(\left(\left(\dfrac{54}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{4}\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{75}\right)+\left(\left(-\dfrac{41}{2}\right)\sqrt{9}\right)+2-\left(\left(\dfrac{59}{4}\right)\sqrt{75}\right)+\left(-\dfrac{33}{5}\right)\sqrt{75}+\left(\dfrac{64}{3}\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{213}{8}\right)\sqrt{3}\right)+\left(\left(\left(\dfrac{108}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{3}\right)-\dfrac{123}{2}+2-\left(\left(\dfrac{295}{4}\right)\sqrt{3}\right)+\left(-33\right)\sqrt{3}+64\right)\\
&=&\left(\dfrac{213}{8}\right)\sqrt{3}+\left(\left(\dfrac{108}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{3}\right)-\dfrac{123}{2}+2-\left(\left(\dfrac{295}{4}\right)\sqrt{3}\right)+\left(-33\right)\sqrt{3}+64\\
&=&\left(-\dfrac{2341}{40}\right)\sqrt{3}+\dfrac{9}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{71}{8}\right)\sqrt{27}\right)-\left(\left(\left(\dfrac{54}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{4}\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{75}\right)+\left(\left(-\dfrac{41}{2}\right)\sqrt{9}\right)+2-\left(\left(\dfrac{59}{4}\right)\sqrt{75}\right)+\left(-\dfrac{33}{5}\right)\sqrt{75}+\left(\dfrac{64}{3}\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{213}{8}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{108}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{3}\right)-\dfrac{123}{2}+2-\left(\left(\dfrac{295}{4}\right)\sqrt{3}\right)+\left(-33\right)\sqrt{3}+64\right)\\
&=&\left(\left(\dfrac{213}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{1703}{20}\right)\sqrt{3}+\dfrac{9}{2}\right)\\
&=&\left(\dfrac{213}{8}\right)\sqrt{3}+\left(\dfrac{1703}{20}\right)\sqrt{3}-\dfrac{9}{2}\\
&=&\left(\dfrac{4471}{40}\right)\sqrt{3}-\dfrac{9}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{71}{8}\right)\sqrt{27}\right)\times\left(\left(\left(\dfrac{54}{5}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{4}\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{75}\right)+\left(\left(-\dfrac{41}{2}\right)\sqrt{9}\right)+2-\left(\left(\dfrac{59}{4}\right)\sqrt{75}\right)+\left(-\dfrac{33}{5}\right)\sqrt{75}+\left(\dfrac{64}{3}\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{213}{8}\right)\sqrt{3}\right)\times\left(\left(\left(\dfrac{108}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{3}\right)-\dfrac{123}{2}+2-\left(\left(\dfrac{295}{4}\right)\sqrt{3}\right)+\left(-33\right)\sqrt{3}+64\right)\\
&=&\left(\left(\dfrac{213}{8}\right)\sqrt{3}\right)\left(\left(-\dfrac{1703}{20}\right)\sqrt{3}+\dfrac{9}{2}\right)\\
&=&\left(-\dfrac{362739}{160}\right)\sqrt{9}+\left(\dfrac{1917}{16}\right)\sqrt{3}\\
&=&-\dfrac{1088217}{160}+\left(\dfrac{1917}{16}\right)\sqrt{3}\\
\end{eqnarray*}