L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{41}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{27}\right)-\left(\left(4\right)\sqrt{27}\right)-\left(\left(\dfrac{23}{8}\right)\sqrt{12}+\left(\dfrac{11}{7}\right)\sqrt{27}+\dfrac{67}{5}+\left(0\right)\sqrt{12}+\left(\dfrac{17}{3}\right)\sqrt{27}\right)\) et \( Y=\left(\dfrac{32}{9}\right)\sqrt{9}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\dfrac{41}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{27}\right)-\left(\left(4\right)\sqrt{27}\right)-\left(\left(\dfrac{23}{8}\right)\sqrt{12}+\left(\dfrac{11}{7}\right)\sqrt{27}+\dfrac{67}{5}+\left(0\right)\sqrt{12}+\left(\dfrac{17}{3}\right)\sqrt{27}\right)\right)+\left(\left(\dfrac{32}{9}\right)\sqrt{9}\right)\\
&=&\left(\dfrac{123}{2}-\left(\left(-\dfrac{55}{3}\right)\sqrt{3}\right)-\left(\left(12\right)\sqrt{3}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{3}+\left(\dfrac{33}{7}\right)\sqrt{3}+\dfrac{67}{5}+\left(0\right)\sqrt{3}+\left(17\right)\sqrt{3}\right)\right)+\left(\dfrac{32}{3}\right)\\
&=&\dfrac{123}{2}-\left(\left(-\dfrac{55}{3}\right)\sqrt{3}\right)-\left(\left(12\right)\sqrt{3}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{3}+\left(\dfrac{33}{7}\right)\sqrt{3}+\dfrac{67}{5}+\left(0\right)\sqrt{3}+\left(17\right)\sqrt{3}\right)+\dfrac{32}{3}\\
&=&\dfrac{1763}{30}+\left(-\dfrac{1775}{84}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{41}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{27}\right)-\left(\left(4\right)\sqrt{27}\right)-\left(\left(\dfrac{23}{8}\right)\sqrt{12}+\left(\dfrac{11}{7}\right)\sqrt{27}+\dfrac{67}{5}+\left(0\right)\sqrt{12}+\left(\dfrac{17}{3}\right)\sqrt{27}\right)\right)-\left(\left(\dfrac{32}{9}\right)\sqrt{9}\right)\\
&=&\left(\dfrac{123}{2}-\left(\left(-\dfrac{55}{3}\right)\sqrt{3}\right)-\left(\left(12\right)\sqrt{3}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{3}+\left(\dfrac{33}{7}\right)\sqrt{3}+\dfrac{67}{5}+\left(0\right)\sqrt{3}+\left(17\right)\sqrt{3}\right)\right)-\left(\dfrac{32}{3}\right)\\
&=&\left(\dfrac{481}{10}+\left(-\dfrac{1775}{84}\right)\sqrt{3}\right)-\left(\dfrac{32}{3}\right)\\
&=&\dfrac{481}{10}+\left(-\dfrac{1775}{84}\right)\sqrt{3}+-\dfrac{32}{3}\\
&=&\dfrac{1123}{30}+\left(-\dfrac{1775}{84}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{41}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{27}\right)-\left(\left(4\right)\sqrt{27}\right)-\left(\left(\dfrac{23}{8}\right)\sqrt{12}+\left(\dfrac{11}{7}\right)\sqrt{27}+\dfrac{67}{5}+\left(0\right)\sqrt{12}+\left(\dfrac{17}{3}\right)\sqrt{27}\right)\right)\times\left(\left(\dfrac{32}{9}\right)\sqrt{9}\right)\\
&=&\left(\dfrac{123}{2}-\left(\left(-\dfrac{55}{3}\right)\sqrt{3}\right)-\left(\left(12\right)\sqrt{3}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{3}+\left(\dfrac{33}{7}\right)\sqrt{3}+\dfrac{67}{5}+\left(0\right)\sqrt{3}+\left(17\right)\sqrt{3}\right)\right)\times\left(\dfrac{32}{3}\right)\\
&=&\left(\dfrac{481}{10}+\left(-\dfrac{1775}{84}\right)\sqrt{3}\right)\left(\dfrac{32}{3}\right)\\
&=&\dfrac{7696}{15}+\left(-\dfrac{14200}{63}\right)\sqrt{3}\\
&=&\dfrac{7696}{15}+\left(-\dfrac{14200}{63}\right)\sqrt{3}\\
\end{eqnarray*}