L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{69}{5}\right)\sqrt{18}+\left(\dfrac{4}{3}\right)\sqrt{4}-\dfrac{10}{3}-\dfrac{28}{3}+\left(-4\right)\sqrt{50}+\left(-\dfrac{19}{4}\right)\sqrt{8}+\left(-\dfrac{19}{4}\right)\sqrt{8}\) et \( Y=\left(-\dfrac{16}{3}\right)\sqrt{18}\) . 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{69}{5}\right)\sqrt{18}+\left(\dfrac{4}{3}\right)\sqrt{4}-\dfrac{10}{3}-\dfrac{28}{3}+\left(-4\right)\sqrt{50}+\left(-\dfrac{19}{4}\right)\sqrt{8}+\left(-\dfrac{19}{4}\right)\sqrt{8}\right)+\left(\left(-\dfrac{16}{3}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{207}{5}\right)\sqrt{2}+\dfrac{8}{3}-\dfrac{10}{3}-\dfrac{28}{3}+\left(-20\right)\sqrt{2}+\left(-\dfrac{19}{2}\right)\sqrt{2}+\left(-\dfrac{19}{2}\right)\sqrt{2}\right)+\left(\left(-16\right)\sqrt{2}\right)\\
&=&\left(\dfrac{207}{5}\right)\sqrt{2}+\dfrac{8}{3}-\dfrac{10}{3}-\dfrac{28}{3}+\left(-20\right)\sqrt{2}+\left(-\dfrac{19}{2}\right)\sqrt{2}+\left(-\dfrac{19}{2}\right)\sqrt{2}+\left(-16\right)\sqrt{2}\\
&=&\left(-\dfrac{68}{5}\right)\sqrt{2}-10\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{69}{5}\right)\sqrt{18}+\left(\dfrac{4}{3}\right)\sqrt{4}-\dfrac{10}{3}-\dfrac{28}{3}+\left(-4\right)\sqrt{50}+\left(-\dfrac{19}{4}\right)\sqrt{8}+\left(-\dfrac{19}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{16}{3}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{207}{5}\right)\sqrt{2}+\dfrac{8}{3}-\dfrac{10}{3}-\dfrac{28}{3}+\left(-20\right)\sqrt{2}+\left(-\dfrac{19}{2}\right)\sqrt{2}+\left(-\dfrac{19}{2}\right)\sqrt{2}\right)-\left(\left(-16\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{12}{5}\right)\sqrt{2}-10\right)-\left(\left(-16\right)\sqrt{2}\right)\\
&=&\left(\dfrac{12}{5}\right)\sqrt{2}-10+\left(16\right)\sqrt{2}\\
&=&\left(\dfrac{92}{5}\right)\sqrt{2}-10\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{69}{5}\right)\sqrt{18}+\left(\dfrac{4}{3}\right)\sqrt{4}-\dfrac{10}{3}-\dfrac{28}{3}+\left(-4\right)\sqrt{50}+\left(-\dfrac{19}{4}\right)\sqrt{8}+\left(-\dfrac{19}{4}\right)\sqrt{8}\right)\times\left(\left(-\dfrac{16}{3}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{207}{5}\right)\sqrt{2}+\dfrac{8}{3}-\dfrac{10}{3}-\dfrac{28}{3}+\left(-20\right)\sqrt{2}+\left(-\dfrac{19}{2}\right)\sqrt{2}+\left(-\dfrac{19}{2}\right)\sqrt{2}\right)\times\left(\left(-16\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{12}{5}\right)\sqrt{2}-10\right)\left(\left(-16\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{192}{5}\right)\sqrt{4}+\left(160\right)\sqrt{2}\\
&=&-\dfrac{384}{5}+\left(160\right)\sqrt{2}\\
\end{eqnarray*}