L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{5}{9}\right)\sqrt{8}\) et \( Y=\left(\dfrac{79}{4}\right)\sqrt{8}+\left(-\dfrac{8}{7}\right)\sqrt{50}+\left(\dfrac{9}{5}\right)\sqrt{18}+\left(\left(-9\right)\sqrt{18}\right)+6-\left(\left(-\dfrac{26}{3}\right)\sqrt{4}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{5}{9}\right)\sqrt{8}\right)+\left(\left(\dfrac{79}{4}\right)\sqrt{8}+\left(-\dfrac{8}{7}\right)\sqrt{50}+\left(\dfrac{9}{5}\right)\sqrt{18}+\left(\left(-9\right)\sqrt{18}\right)+6-\left(\left(-\dfrac{26}{3}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\dfrac{10}{9}\right)\sqrt{2}\right)+\left(\left(\dfrac{79}{2}\right)\sqrt{2}+\left(-\dfrac{40}{7}\right)\sqrt{2}+\left(\dfrac{27}{5}\right)\sqrt{2}+\left(\left(-27\right)\sqrt{2}\right)+6+\dfrac{52}{3}\right)\\
&=&\left(\dfrac{10}{9}\right)\sqrt{2}+\left(\dfrac{79}{2}\right)\sqrt{2}+\left(-\dfrac{40}{7}\right)\sqrt{2}+\left(\dfrac{27}{5}\right)\sqrt{2}+\left(\left(-27\right)\sqrt{2}\right)+6+\dfrac{52}{3}\\
&=&\left(\dfrac{8377}{630}\right)\sqrt{2}+\dfrac{70}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{5}{9}\right)\sqrt{8}\right)-\left(\left(\dfrac{79}{4}\right)\sqrt{8}+\left(-\dfrac{8}{7}\right)\sqrt{50}+\left(\dfrac{9}{5}\right)\sqrt{18}+\left(\left(-9\right)\sqrt{18}\right)+6-\left(\left(-\dfrac{26}{3}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\dfrac{10}{9}\right)\sqrt{2}\right)-\left(\left(\dfrac{79}{2}\right)\sqrt{2}+\left(-\dfrac{40}{7}\right)\sqrt{2}+\left(\dfrac{27}{5}\right)\sqrt{2}+\left(\left(-27\right)\sqrt{2}\right)+6+\dfrac{52}{3}\right)\\
&=&\left(\left(\dfrac{10}{9}\right)\sqrt{2}\right)-\left(\left(\dfrac{853}{70}\right)\sqrt{2}+\dfrac{70}{3}\right)\\
&=&\left(\dfrac{10}{9}\right)\sqrt{2}+\left(-\dfrac{853}{70}\right)\sqrt{2}-\dfrac{70}{3}\\
&=&\left(-\dfrac{6977}{630}\right)\sqrt{2}-\dfrac{70}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{5}{9}\right)\sqrt{8}\right)\times\left(\left(\dfrac{79}{4}\right)\sqrt{8}+\left(-\dfrac{8}{7}\right)\sqrt{50}+\left(\dfrac{9}{5}\right)\sqrt{18}+\left(\left(-9\right)\sqrt{18}\right)+6-\left(\left(-\dfrac{26}{3}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\dfrac{10}{9}\right)\sqrt{2}\right)\times\left(\left(\dfrac{79}{2}\right)\sqrt{2}+\left(-\dfrac{40}{7}\right)\sqrt{2}+\left(\dfrac{27}{5}\right)\sqrt{2}+\left(\left(-27\right)\sqrt{2}\right)+6+\dfrac{52}{3}\right)\\
&=&\left(\left(\dfrac{10}{9}\right)\sqrt{2}\right)\left(\left(\dfrac{853}{70}\right)\sqrt{2}+\dfrac{70}{3}\right)\\
&=&\left(\dfrac{853}{63}\right)\sqrt{4}+\left(\dfrac{700}{27}\right)\sqrt{2}\\
&=&\dfrac{1706}{63}+\left(\dfrac{700}{27}\right)\sqrt{2}\\
\end{eqnarray*}