L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\left(\dfrac{17}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{2}{5}\right)\sqrt{45}\right)\right)-\left(\dfrac{29}{3}+\left(-\dfrac{15}{7}\right)\sqrt{45}+\left(\dfrac{66}{7}\right)\sqrt{25}\right)\) et \( Y=\left(\left(6\right)\sqrt{20}\right)-\left(\left(\left(\dfrac{77}{3}\right)\sqrt{20}\right)-\left(\left(-9\right)\sqrt{125}\right)-\left(\left(-\dfrac{58}{9}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{20}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(\dfrac{17}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{2}{5}\right)\sqrt{45}\right)\right)-\left(\dfrac{29}{3}+\left(-\dfrac{15}{7}\right)\sqrt{45}+\left(\dfrac{66}{7}\right)\sqrt{25}\right)\right)+\left(\left(\left(6\right)\sqrt{20}\right)-\left(\left(\left(\dfrac{77}{3}\right)\sqrt{20}\right)-\left(\left(-9\right)\sqrt{125}\right)-\left(\left(-\dfrac{58}{9}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{17}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{6}{5}\right)\sqrt{5}\right)\right)-\left(\dfrac{29}{3}+\left(-\dfrac{45}{7}\right)\sqrt{5}+\dfrac{330}{7}\right)\right)+\left(\left(\left(12\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{154}{3}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)-\left(\left(-\dfrac{290}{9}\right)\sqrt{5}\right)\right)-\left(\left(9\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\left(\dfrac{17}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{6}{5}\right)\sqrt{5}\right)\right)-\left(\dfrac{29}{3}+\left(-\dfrac{45}{7}\right)\sqrt{5}+\dfrac{330}{7}\right)+\left(\left(12\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{154}{3}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)-\left(\left(-\dfrac{290}{9}\right)\sqrt{5}\right)\right)-\left(\left(9\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{68939}{630}\right)\sqrt{5}-\dfrac{1193}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{17}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{2}{5}\right)\sqrt{45}\right)\right)-\left(\dfrac{29}{3}+\left(-\dfrac{15}{7}\right)\sqrt{45}+\left(\dfrac{66}{7}\right)\sqrt{25}\right)\right)-\left(\left(\left(6\right)\sqrt{20}\right)-\left(\left(\left(\dfrac{77}{3}\right)\sqrt{20}\right)-\left(\left(-9\right)\sqrt{125}\right)-\left(\left(-\dfrac{58}{9}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{17}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{6}{5}\right)\sqrt{5}\right)\right)-\left(\dfrac{29}{3}+\left(-\dfrac{45}{7}\right)\sqrt{5}+\dfrac{330}{7}\right)\right)-\left(\left(\left(12\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{154}{3}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)-\left(\left(-\dfrac{290}{9}\right)\sqrt{5}\right)\right)-\left(\left(9\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{1129}{70}\right)\sqrt{5}-\dfrac{1193}{21}\right)-\left(\left(-\dfrac{1130}{9}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{1129}{70}\right)\sqrt{5}-\dfrac{1193}{21}+\left(\dfrac{1130}{9}\right)\sqrt{5}\\
&=&\left(\dfrac{89261}{630}\right)\sqrt{5}-\dfrac{1193}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{17}{6}\right)\sqrt{45}\right)-\left(\left(-\dfrac{2}{5}\right)\sqrt{45}\right)\right)-\left(\dfrac{29}{3}+\left(-\dfrac{15}{7}\right)\sqrt{45}+\left(\dfrac{66}{7}\right)\sqrt{25}\right)\right)\times\left(\left(\left(6\right)\sqrt{20}\right)-\left(\left(\left(\dfrac{77}{3}\right)\sqrt{20}\right)-\left(\left(-9\right)\sqrt{125}\right)-\left(\left(-\dfrac{58}{9}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{20}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{17}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{6}{5}\right)\sqrt{5}\right)\right)-\left(\dfrac{29}{3}+\left(-\dfrac{45}{7}\right)\sqrt{5}+\dfrac{330}{7}\right)\right)\times\left(\left(\left(12\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{154}{3}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)-\left(\left(-\dfrac{290}{9}\right)\sqrt{5}\right)\right)-\left(\left(9\right)\sqrt{5}\right)\right)\\
&=&\left(\left(\dfrac{1129}{70}\right)\sqrt{5}-\dfrac{1193}{21}\right)\left(\left(-\dfrac{1130}{9}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{127577}{63}\right)\sqrt{25}+\left(\dfrac{1348090}{189}\right)\sqrt{5}\\
&=&-\dfrac{637885}{63}+\left(\dfrac{1348090}{189}\right)\sqrt{5}\\
\end{eqnarray*}