L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{47}{3}\right)\sqrt{125}+\left(6\right)\sqrt{45}+\left(\dfrac{39}{5}\right)\sqrt{125}-\dfrac{21}{2}-\left(\left(-4\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{45}\right)\) et \( Y=\left(-\dfrac{10}{3}\right)\sqrt{25}+\left(-\dfrac{49}{9}\right)\sqrt{25}+\left(\dfrac{57}{8}\right)\sqrt{25}+\left(-\dfrac{39}{4}\right)\sqrt{20}+\left(-\dfrac{32}{7}\right)\sqrt{125}+\left(\dfrac{67}{9}\right)\sqrt{20}\) . 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{47}{3}\right)\sqrt{125}+\left(6\right)\sqrt{45}+\left(\dfrac{39}{5}\right)\sqrt{125}-\dfrac{21}{2}-\left(\left(-4\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{45}\right)\right)+\left(\left(-\dfrac{10}{3}\right)\sqrt{25}+\left(-\dfrac{49}{9}\right)\sqrt{25}+\left(\dfrac{57}{8}\right)\sqrt{25}+\left(-\dfrac{39}{4}\right)\sqrt{20}+\left(-\dfrac{32}{7}\right)\sqrt{125}+\left(\dfrac{67}{9}\right)\sqrt{20}\right)\\
&=&\left(\left(-\dfrac{235}{3}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\left(39\right)\sqrt{5}-\dfrac{21}{2}-\left(\left(-8\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)\right)+\left(-\dfrac{50}{3}-\dfrac{245}{9}+\dfrac{285}{8}+\left(-\dfrac{39}{2}\right)\sqrt{5}+\left(-\dfrac{160}{7}\right)\sqrt{5}+\left(\dfrac{134}{9}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{235}{3}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\left(39\right)\sqrt{5}-\dfrac{21}{2}-\left(\left(-8\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)-\dfrac{50}{3}-\dfrac{245}{9}+\dfrac{285}{8}+\left(-\dfrac{39}{2}\right)\sqrt{5}+\left(-\dfrac{160}{7}\right)\sqrt{5}+\left(\dfrac{134}{9}\right)\sqrt{5}\\
&=&\left(-\dfrac{7409}{126}\right)\sqrt{5}-\dfrac{1351}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{47}{3}\right)\sqrt{125}+\left(6\right)\sqrt{45}+\left(\dfrac{39}{5}\right)\sqrt{125}-\dfrac{21}{2}-\left(\left(-4\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{45}\right)\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{25}+\left(-\dfrac{49}{9}\right)\sqrt{25}+\left(\dfrac{57}{8}\right)\sqrt{25}+\left(-\dfrac{39}{4}\right)\sqrt{20}+\left(-\dfrac{32}{7}\right)\sqrt{125}+\left(\dfrac{67}{9}\right)\sqrt{20}\right)\\
&=&\left(\left(-\dfrac{235}{3}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\left(39\right)\sqrt{5}-\dfrac{21}{2}-\left(\left(-8\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)\right)-\left(-\dfrac{50}{3}-\dfrac{245}{9}+\dfrac{285}{8}+\left(-\dfrac{39}{2}\right)\sqrt{5}+\left(-\dfrac{160}{7}\right)\sqrt{5}+\left(\dfrac{134}{9}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{94}{3}\right)\sqrt{5}-\dfrac{21}{2}\right)-\left(-\dfrac{595}{72}+\left(-\dfrac{3461}{126}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{94}{3}\right)\sqrt{5}-\dfrac{21}{2}+\dfrac{595}{72}+\left(\dfrac{3461}{126}\right)\sqrt{5}\\
&=&\left(-\dfrac{487}{126}\right)\sqrt{5}-\dfrac{161}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{47}{3}\right)\sqrt{125}+\left(6\right)\sqrt{45}+\left(\dfrac{39}{5}\right)\sqrt{125}-\dfrac{21}{2}-\left(\left(-4\right)\sqrt{20}\right)-\left(\left(6\right)\sqrt{45}\right)\right)\times\left(\left(-\dfrac{10}{3}\right)\sqrt{25}+\left(-\dfrac{49}{9}\right)\sqrt{25}+\left(\dfrac{57}{8}\right)\sqrt{25}+\left(-\dfrac{39}{4}\right)\sqrt{20}+\left(-\dfrac{32}{7}\right)\sqrt{125}+\left(\dfrac{67}{9}\right)\sqrt{20}\right)\\
&=&\left(\left(-\dfrac{235}{3}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\left(39\right)\sqrt{5}-\dfrac{21}{2}-\left(\left(-8\right)\sqrt{5}\right)-\left(\left(18\right)\sqrt{5}\right)\right)\times\left(-\dfrac{50}{3}-\dfrac{245}{9}+\dfrac{285}{8}+\left(-\dfrac{39}{2}\right)\sqrt{5}+\left(-\dfrac{160}{7}\right)\sqrt{5}+\left(\dfrac{134}{9}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{94}{3}\right)\sqrt{5}-\dfrac{21}{2}\right)\left(-\dfrac{595}{72}+\left(-\dfrac{3461}{126}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{29557}{54}\right)\sqrt{5}+\left(\dfrac{162667}{189}\right)\sqrt{25}+\dfrac{4165}{48}\\
&=&\left(\dfrac{29557}{54}\right)\sqrt{5}+\dfrac{13275755}{3024}\\
\end{eqnarray*}