L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{35}{3}\right)\sqrt{8}\) et \( Y=\left(\left(\left(-8\right)\sqrt{18}\right)-\left(\left(\dfrac{9}{8}\right)\sqrt{8}\right)\right)-\left(\left(\left(-\dfrac{21}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{9}{8}\right)\sqrt{8}\right)\right)-\left(\left(2\right)\sqrt{8}+\left(\dfrac{18}{5}\right)\sqrt{4}\right)\) . 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{35}{3}\right)\sqrt{8}\right)+\left(\left(\left(\left(-8\right)\sqrt{18}\right)-\left(\left(\dfrac{9}{8}\right)\sqrt{8}\right)\right)-\left(\left(\left(-\dfrac{21}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{9}{8}\right)\sqrt{8}\right)\right)-\left(\left(2\right)\sqrt{8}+\left(\dfrac{18}{5}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\dfrac{70}{3}\right)\sqrt{2}\right)+\left(\left(\left(\left(-24\right)\sqrt{2}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{42}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{2}\right)\right)-\left(\left(4\right)\sqrt{2}+\dfrac{36}{5}\right)\right)\\
&=&\left(\dfrac{70}{3}\right)\sqrt{2}+\left(\left(\left(-24\right)\sqrt{2}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{42}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{2}\right)\right)-\left(\left(4\right)\sqrt{2}+\dfrac{36}{5}\right)\\
&=&\left(\dfrac{899}{60}\right)\sqrt{2}-\dfrac{36}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{35}{3}\right)\sqrt{8}\right)-\left(\left(\left(\left(-8\right)\sqrt{18}\right)-\left(\left(\dfrac{9}{8}\right)\sqrt{8}\right)\right)-\left(\left(\left(-\dfrac{21}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{9}{8}\right)\sqrt{8}\right)\right)-\left(\left(2\right)\sqrt{8}+\left(\dfrac{18}{5}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\dfrac{70}{3}\right)\sqrt{2}\right)-\left(\left(\left(\left(-24\right)\sqrt{2}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{42}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{2}\right)\right)-\left(\left(4\right)\sqrt{2}+\dfrac{36}{5}\right)\right)\\
&=&\left(\left(\dfrac{70}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{167}{20}\right)\sqrt{2}-\dfrac{36}{5}\right)\\
&=&\left(\dfrac{70}{3}\right)\sqrt{2}+\left(\dfrac{167}{20}\right)\sqrt{2}+\dfrac{36}{5}\\
&=&\left(\dfrac{1901}{60}\right)\sqrt{2}+\dfrac{36}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{35}{3}\right)\sqrt{8}\right)\times\left(\left(\left(\left(-8\right)\sqrt{18}\right)-\left(\left(\dfrac{9}{8}\right)\sqrt{8}\right)\right)-\left(\left(\left(-\dfrac{21}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{9}{8}\right)\sqrt{8}\right)\right)-\left(\left(2\right)\sqrt{8}+\left(\dfrac{18}{5}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\dfrac{70}{3}\right)\sqrt{2}\right)\times\left(\left(\left(\left(-24\right)\sqrt{2}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{42}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{2}\right)\right)-\left(\left(4\right)\sqrt{2}+\dfrac{36}{5}\right)\right)\\
&=&\left(\left(\dfrac{70}{3}\right)\sqrt{2}\right)\left(\left(-\dfrac{167}{20}\right)\sqrt{2}-\dfrac{36}{5}\right)\\
&=&\left(-\dfrac{1169}{6}\right)\sqrt{4}+\left(-168\right)\sqrt{2}\\
&=&-\dfrac{1169}{3}+\left(-168\right)\sqrt{2}\\
\end{eqnarray*}