L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{74}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{17}{5}\right)\sqrt{50}\right)\) et \( Y=\left(-\dfrac{73}{9}+\left(-6\right)\sqrt{50}+\left(\dfrac{37}{6}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{74}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{17}{5}\right)\sqrt{50}\right)\right)+\left(\left(-\dfrac{73}{9}+\left(-6\right)\sqrt{50}+\left(\dfrac{37}{6}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)\right)\\
&=&\left(-\dfrac{148}{9}-\left(\left(-17\right)\sqrt{2}\right)\right)+\left(\left(-\dfrac{73}{9}+\left(-30\right)\sqrt{2}+\dfrac{37}{3}\right)-\left(\left(\dfrac{33}{2}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\right)\\
&=&-\dfrac{148}{9}-\left(\left(-17\right)\sqrt{2}\right)+\left(-\dfrac{73}{9}+\left(-30\right)\sqrt{2}+\dfrac{37}{3}\right)-\left(\left(\dfrac{33}{2}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\\
&=&-\dfrac{110}{9}+\left(-\dfrac{59}{2}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{74}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{17}{5}\right)\sqrt{50}\right)\right)-\left(\left(-\dfrac{73}{9}+\left(-6\right)\sqrt{50}+\left(\dfrac{37}{6}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)\right)\\
&=&\left(-\dfrac{148}{9}-\left(\left(-17\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{73}{9}+\left(-30\right)\sqrt{2}+\dfrac{37}{3}\right)-\left(\left(\dfrac{33}{2}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\right)\\
&=&\left(-\dfrac{148}{9}+\left(17\right)\sqrt{2}\right)-\left(\dfrac{38}{9}+\left(-\dfrac{93}{2}\right)\sqrt{2}\right)\\
&=&-\dfrac{148}{9}+\left(17\right)\sqrt{2}+-\dfrac{38}{9}+\left(\dfrac{93}{2}\right)\sqrt{2}\\
&=&-\dfrac{62}{3}+\left(\dfrac{127}{2}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{74}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{17}{5}\right)\sqrt{50}\right)\right)\times\left(\left(-\dfrac{73}{9}+\left(-6\right)\sqrt{50}+\left(\dfrac{37}{6}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)\right)\\
&=&\left(-\dfrac{148}{9}-\left(\left(-17\right)\sqrt{2}\right)\right)\times\left(\left(-\dfrac{73}{9}+\left(-30\right)\sqrt{2}+\dfrac{37}{3}\right)-\left(\left(\dfrac{33}{2}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\right)\\
&=&\left(-\dfrac{148}{9}+\left(17\right)\sqrt{2}\right)\left(\dfrac{38}{9}+\left(-\dfrac{93}{2}\right)\sqrt{2}\right)\\
&=&-\dfrac{5624}{81}+\left(\dfrac{7528}{9}\right)\sqrt{2}+\left(-\dfrac{1581}{2}\right)\sqrt{4}\\
&=&-\dfrac{133685}{81}+\left(\dfrac{7528}{9}\right)\sqrt{2}\\
\end{eqnarray*}