L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{59}{8}\right)\sqrt{20}\) et \( Y=\dfrac{9}{4}+\left(6\right)\sqrt{20}+\left(5\right)\sqrt{45}+\left(-\dfrac{17}{4}\right)\sqrt{45}+\left(-\dfrac{33}{5}\right)\sqrt{25}+\left(-7\right)\sqrt{45}+\left(-\dfrac{4}{7}\right)\sqrt{45}+\left(-\dfrac{2}{3}\right)\sqrt{45}+\left(-\dfrac{73}{9}\right)\sqrt{125}\) . 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{59}{8}\right)\sqrt{20}\right)+\left(\dfrac{9}{4}+\left(6\right)\sqrt{20}+\left(5\right)\sqrt{45}+\left(-\dfrac{17}{4}\right)\sqrt{45}+\left(-\dfrac{33}{5}\right)\sqrt{25}+\left(-7\right)\sqrt{45}+\left(-\dfrac{4}{7}\right)\sqrt{45}+\left(-\dfrac{2}{3}\right)\sqrt{45}+\left(-\dfrac{73}{9}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{59}{4}\right)\sqrt{5}\right)+\left(\dfrac{9}{4}+\left(12\right)\sqrt{5}+\left(15\right)\sqrt{5}+\left(-\dfrac{51}{4}\right)\sqrt{5}-33+\left(-21\right)\sqrt{5}+\left(-\dfrac{12}{7}\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\left(-\dfrac{365}{9}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{59}{4}\right)\sqrt{5}+\dfrac{9}{4}+\left(12\right)\sqrt{5}+\left(15\right)\sqrt{5}+\left(-\dfrac{51}{4}\right)\sqrt{5}-33+\left(-21\right)\sqrt{5}+\left(-\dfrac{12}{7}\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\left(-\dfrac{365}{9}\right)\sqrt{5}\\
&=&\left(-\dfrac{8287}{126}\right)\sqrt{5}-\dfrac{123}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{59}{8}\right)\sqrt{20}\right)-\left(\dfrac{9}{4}+\left(6\right)\sqrt{20}+\left(5\right)\sqrt{45}+\left(-\dfrac{17}{4}\right)\sqrt{45}+\left(-\dfrac{33}{5}\right)\sqrt{25}+\left(-7\right)\sqrt{45}+\left(-\dfrac{4}{7}\right)\sqrt{45}+\left(-\dfrac{2}{3}\right)\sqrt{45}+\left(-\dfrac{73}{9}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{59}{4}\right)\sqrt{5}\right)-\left(\dfrac{9}{4}+\left(12\right)\sqrt{5}+\left(15\right)\sqrt{5}+\left(-\dfrac{51}{4}\right)\sqrt{5}-33+\left(-21\right)\sqrt{5}+\left(-\dfrac{12}{7}\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\left(-\dfrac{365}{9}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{59}{4}\right)\sqrt{5}\right)-\left(-\dfrac{123}{4}+\left(-\dfrac{12857}{252}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{59}{4}\right)\sqrt{5}+\dfrac{123}{4}+\left(\dfrac{12857}{252}\right)\sqrt{5}\\
&=&\left(\dfrac{2285}{63}\right)\sqrt{5}+\dfrac{123}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{59}{8}\right)\sqrt{20}\right)\times\left(\dfrac{9}{4}+\left(6\right)\sqrt{20}+\left(5\right)\sqrt{45}+\left(-\dfrac{17}{4}\right)\sqrt{45}+\left(-\dfrac{33}{5}\right)\sqrt{25}+\left(-7\right)\sqrt{45}+\left(-\dfrac{4}{7}\right)\sqrt{45}+\left(-\dfrac{2}{3}\right)\sqrt{45}+\left(-\dfrac{73}{9}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{59}{4}\right)\sqrt{5}\right)\times\left(\dfrac{9}{4}+\left(12\right)\sqrt{5}+\left(15\right)\sqrt{5}+\left(-\dfrac{51}{4}\right)\sqrt{5}-33+\left(-21\right)\sqrt{5}+\left(-\dfrac{12}{7}\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\left(-\dfrac{365}{9}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{59}{4}\right)\sqrt{5}\right)\left(-\dfrac{123}{4}+\left(-\dfrac{12857}{252}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{7257}{16}\right)\sqrt{5}+\left(\dfrac{758563}{1008}\right)\sqrt{25}\\
&=&\left(\dfrac{7257}{16}\right)\sqrt{5}+\dfrac{3792815}{1008}\\
\end{eqnarray*}