L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-3\right)\sqrt{50}+\left(\dfrac{37}{8}\right)\sqrt{50}\) et \( Y=\left(-\dfrac{37}{7}\right)\sqrt{50}+\left(\left(\dfrac{20}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{4}\right)-\dfrac{1}{4}-\dfrac{11}{3}-\left(\left(5\right)\sqrt{18}\right)+\left(\dfrac{37}{2}\right)\sqrt{18}+\left(-\dfrac{43}{9}\right)\sqrt{8}+\dfrac{1}{4}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-3\right)\sqrt{50}+\left(\dfrac{37}{8}\right)\sqrt{50}\right)+\left(\left(-\dfrac{37}{7}\right)\sqrt{50}+\left(\left(\dfrac{20}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{4}\right)-\dfrac{1}{4}-\dfrac{11}{3}-\left(\left(5\right)\sqrt{18}\right)+\left(\dfrac{37}{2}\right)\sqrt{18}+\left(-\dfrac{43}{9}\right)\sqrt{8}+\dfrac{1}{4}\right)\\
&=&\left(\left(-15\right)\sqrt{2}+\left(\dfrac{185}{8}\right)\sqrt{2}\right)+\left(\left(-\dfrac{185}{7}\right)\sqrt{2}+\left(\left(\dfrac{60}{7}\right)\sqrt{2}\right)+\dfrac{58}{3}-\dfrac{1}{4}-\dfrac{11}{3}-\left(\left(15\right)\sqrt{2}\right)+\left(\dfrac{111}{2}\right)\sqrt{2}+\left(-\dfrac{86}{9}\right)\sqrt{2}+\dfrac{1}{4}\right)\\
&=&\left(-15\right)\sqrt{2}+\left(\dfrac{185}{8}\right)\sqrt{2}+\left(-\dfrac{185}{7}\right)\sqrt{2}+\left(\left(\dfrac{60}{7}\right)\sqrt{2}\right)+\dfrac{58}{3}-\dfrac{1}{4}-\dfrac{11}{3}-\left(\left(15\right)\sqrt{2}\right)+\left(\dfrac{111}{2}\right)\sqrt{2}+\left(-\dfrac{86}{9}\right)\sqrt{2}+\dfrac{1}{4}\\
&=&\left(\dfrac{10691}{504}\right)\sqrt{2}+\dfrac{47}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-3\right)\sqrt{50}+\left(\dfrac{37}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{37}{7}\right)\sqrt{50}+\left(\left(\dfrac{20}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{4}\right)-\dfrac{1}{4}-\dfrac{11}{3}-\left(\left(5\right)\sqrt{18}\right)+\left(\dfrac{37}{2}\right)\sqrt{18}+\left(-\dfrac{43}{9}\right)\sqrt{8}+\dfrac{1}{4}\right)\\
&=&\left(\left(-15\right)\sqrt{2}+\left(\dfrac{185}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{185}{7}\right)\sqrt{2}+\left(\left(\dfrac{60}{7}\right)\sqrt{2}\right)+\dfrac{58}{3}-\dfrac{1}{4}-\dfrac{11}{3}-\left(\left(15\right)\sqrt{2}\right)+\left(\dfrac{111}{2}\right)\sqrt{2}+\left(-\dfrac{86}{9}\right)\sqrt{2}+\dfrac{1}{4}\right)\\
&=&\left(\left(\dfrac{65}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{1649}{126}\right)\sqrt{2}+\dfrac{47}{3}\right)\\
&=&\left(\dfrac{65}{8}\right)\sqrt{2}+\left(-\dfrac{1649}{126}\right)\sqrt{2}-\dfrac{47}{3}\\
&=&\left(-\dfrac{2501}{504}\right)\sqrt{2}-\dfrac{47}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-3\right)\sqrt{50}+\left(\dfrac{37}{8}\right)\sqrt{50}\right)\times\left(\left(-\dfrac{37}{7}\right)\sqrt{50}+\left(\left(\dfrac{20}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{4}\right)-\dfrac{1}{4}-\dfrac{11}{3}-\left(\left(5\right)\sqrt{18}\right)+\left(\dfrac{37}{2}\right)\sqrt{18}+\left(-\dfrac{43}{9}\right)\sqrt{8}+\dfrac{1}{4}\right)\\
&=&\left(\left(-15\right)\sqrt{2}+\left(\dfrac{185}{8}\right)\sqrt{2}\right)\times\left(\left(-\dfrac{185}{7}\right)\sqrt{2}+\left(\left(\dfrac{60}{7}\right)\sqrt{2}\right)+\dfrac{58}{3}-\dfrac{1}{4}-\dfrac{11}{3}-\left(\left(15\right)\sqrt{2}\right)+\left(\dfrac{111}{2}\right)\sqrt{2}+\left(-\dfrac{86}{9}\right)\sqrt{2}+\dfrac{1}{4}\right)\\
&=&\left(\left(\dfrac{65}{8}\right)\sqrt{2}\right)\left(\left(\dfrac{1649}{126}\right)\sqrt{2}+\dfrac{47}{3}\right)\\
&=&\left(\dfrac{107185}{1008}\right)\sqrt{4}+\left(\dfrac{3055}{24}\right)\sqrt{2}\\
&=&\dfrac{107185}{504}+\left(\dfrac{3055}{24}\right)\sqrt{2}\\
\end{eqnarray*}