L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{29}{3}\right)\sqrt{20}\) et \( Y=\left(-\dfrac{17}{3}\right)\sqrt{25}+\dfrac{63}{8}+\left(-\dfrac{8}{7}\right)\sqrt{25}+\left(-1\right)\sqrt{45}+\left(-\dfrac{38}{3}\right)\sqrt{125}+\left(\dfrac{13}{2}\right)\sqrt{125}+\left(0\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{45}+\left(-\dfrac{31}{6}\right)\sqrt{125}\) . 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{29}{3}\right)\sqrt{20}\right)+\left(\left(-\dfrac{17}{3}\right)\sqrt{25}+\dfrac{63}{8}+\left(-\dfrac{8}{7}\right)\sqrt{25}+\left(-1\right)\sqrt{45}+\left(-\dfrac{38}{3}\right)\sqrt{125}+\left(\dfrac{13}{2}\right)\sqrt{125}+\left(0\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{45}+\left(-\dfrac{31}{6}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{58}{3}\right)\sqrt{5}\right)+\left(-\dfrac{85}{3}+\dfrac{63}{8}-\dfrac{40}{7}+\left(-3\right)\sqrt{5}+\left(-\dfrac{190}{3}\right)\sqrt{5}+\left(\dfrac{65}{2}\right)\sqrt{5}+0+\left(23\right)\sqrt{5}+\left(-\dfrac{155}{6}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{58}{3}\right)\sqrt{5}-\dfrac{85}{3}+\dfrac{63}{8}-\dfrac{40}{7}+\left(-3\right)\sqrt{5}+\left(-\dfrac{190}{3}\right)\sqrt{5}+\left(\dfrac{65}{2}\right)\sqrt{5}+0+\left(23\right)\sqrt{5}+\left(-\dfrac{155}{6}\right)\sqrt{5}\\
&=&\left(-56\right)\sqrt{5}-\dfrac{4397}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{29}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{25}+\dfrac{63}{8}+\left(-\dfrac{8}{7}\right)\sqrt{25}+\left(-1\right)\sqrt{45}+\left(-\dfrac{38}{3}\right)\sqrt{125}+\left(\dfrac{13}{2}\right)\sqrt{125}+\left(0\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{45}+\left(-\dfrac{31}{6}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{58}{3}\right)\sqrt{5}\right)-\left(-\dfrac{85}{3}+\dfrac{63}{8}-\dfrac{40}{7}+\left(-3\right)\sqrt{5}+\left(-\dfrac{190}{3}\right)\sqrt{5}+\left(\dfrac{65}{2}\right)\sqrt{5}+0+\left(23\right)\sqrt{5}+\left(-\dfrac{155}{6}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{58}{3}\right)\sqrt{5}\right)-\left(-\dfrac{4397}{168}+\left(-\dfrac{110}{3}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{58}{3}\right)\sqrt{5}+\dfrac{4397}{168}+\left(\dfrac{110}{3}\right)\sqrt{5}\\
&=&\left(\dfrac{52}{3}\right)\sqrt{5}+\dfrac{4397}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{29}{3}\right)\sqrt{20}\right)\times\left(\left(-\dfrac{17}{3}\right)\sqrt{25}+\dfrac{63}{8}+\left(-\dfrac{8}{7}\right)\sqrt{25}+\left(-1\right)\sqrt{45}+\left(-\dfrac{38}{3}\right)\sqrt{125}+\left(\dfrac{13}{2}\right)\sqrt{125}+\left(0\right)\sqrt{25}+\left(\dfrac{23}{3}\right)\sqrt{45}+\left(-\dfrac{31}{6}\right)\sqrt{125}\right)\\
&=&\left(\left(-\dfrac{58}{3}\right)\sqrt{5}\right)\times\left(-\dfrac{85}{3}+\dfrac{63}{8}-\dfrac{40}{7}+\left(-3\right)\sqrt{5}+\left(-\dfrac{190}{3}\right)\sqrt{5}+\left(\dfrac{65}{2}\right)\sqrt{5}+0+\left(23\right)\sqrt{5}+\left(-\dfrac{155}{6}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{58}{3}\right)\sqrt{5}\right)\left(-\dfrac{4397}{168}+\left(-\dfrac{110}{3}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{127513}{252}\right)\sqrt{5}+\left(\dfrac{6380}{9}\right)\sqrt{25}\\
&=&\left(\dfrac{127513}{252}\right)\sqrt{5}+\dfrac{31900}{9}\\
\end{eqnarray*}