L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{34}{5}\right)\sqrt{125}+\left(\dfrac{19}{3}\right)\sqrt{20}+\left(-\dfrac{7}{2}\right)\sqrt{20}+\left(\dfrac{59}{8}\right)\sqrt{20}-\dfrac{75}{4}-6+\dfrac{38}{3}+\left(-\dfrac{44}{9}\right)\sqrt{45}+\left(\dfrac{31}{9}\right)\sqrt{25}\) et \( Y=\left(\dfrac{61}{3}\right)\sqrt{25}\) . 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{34}{5}\right)\sqrt{125}+\left(\dfrac{19}{3}\right)\sqrt{20}+\left(-\dfrac{7}{2}\right)\sqrt{20}+\left(\dfrac{59}{8}\right)\sqrt{20}-\dfrac{75}{4}-6+\dfrac{38}{3}+\left(-\dfrac{44}{9}\right)\sqrt{45}+\left(\dfrac{31}{9}\right)\sqrt{25}\right)+\left(\left(\dfrac{61}{3}\right)\sqrt{25}\right)\\
&=&\left(\left(-34\right)\sqrt{5}+\left(\dfrac{38}{3}\right)\sqrt{5}+\left(-7\right)\sqrt{5}+\left(\dfrac{59}{4}\right)\sqrt{5}-\dfrac{75}{4}-6+\dfrac{38}{3}+\left(-\dfrac{44}{3}\right)\sqrt{5}+\dfrac{155}{9}\right)+\left(\dfrac{305}{3}\right)\\
&=&\left(-34\right)\sqrt{5}+\left(\dfrac{38}{3}\right)\sqrt{5}+\left(-7\right)\sqrt{5}+\left(\dfrac{59}{4}\right)\sqrt{5}-\dfrac{75}{4}-6+\dfrac{38}{3}+\left(-\dfrac{44}{3}\right)\sqrt{5}+\dfrac{155}{9}+\dfrac{305}{3}\\
&=&\left(-\dfrac{113}{4}\right)\sqrt{5}+\dfrac{3845}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{34}{5}\right)\sqrt{125}+\left(\dfrac{19}{3}\right)\sqrt{20}+\left(-\dfrac{7}{2}\right)\sqrt{20}+\left(\dfrac{59}{8}\right)\sqrt{20}-\dfrac{75}{4}-6+\dfrac{38}{3}+\left(-\dfrac{44}{9}\right)\sqrt{45}+\left(\dfrac{31}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{61}{3}\right)\sqrt{25}\right)\\
&=&\left(\left(-34\right)\sqrt{5}+\left(\dfrac{38}{3}\right)\sqrt{5}+\left(-7\right)\sqrt{5}+\left(\dfrac{59}{4}\right)\sqrt{5}-\dfrac{75}{4}-6+\dfrac{38}{3}+\left(-\dfrac{44}{3}\right)\sqrt{5}+\dfrac{155}{9}\right)-\left(\dfrac{305}{3}\right)\\
&=&\left(\left(-\dfrac{113}{4}\right)\sqrt{5}+\dfrac{185}{36}\right)-\left(\dfrac{305}{3}\right)\\
&=&\left(-\dfrac{113}{4}\right)\sqrt{5}+\dfrac{185}{36}+-\dfrac{305}{3}\\
&=&\left(-\dfrac{113}{4}\right)\sqrt{5}-\dfrac{3475}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{34}{5}\right)\sqrt{125}+\left(\dfrac{19}{3}\right)\sqrt{20}+\left(-\dfrac{7}{2}\right)\sqrt{20}+\left(\dfrac{59}{8}\right)\sqrt{20}-\dfrac{75}{4}-6+\dfrac{38}{3}+\left(-\dfrac{44}{9}\right)\sqrt{45}+\left(\dfrac{31}{9}\right)\sqrt{25}\right)\times\left(\left(\dfrac{61}{3}\right)\sqrt{25}\right)\\
&=&\left(\left(-34\right)\sqrt{5}+\left(\dfrac{38}{3}\right)\sqrt{5}+\left(-7\right)\sqrt{5}+\left(\dfrac{59}{4}\right)\sqrt{5}-\dfrac{75}{4}-6+\dfrac{38}{3}+\left(-\dfrac{44}{3}\right)\sqrt{5}+\dfrac{155}{9}\right)\times\left(\dfrac{305}{3}\right)\\
&=&\left(\left(-\dfrac{113}{4}\right)\sqrt{5}+\dfrac{185}{36}\right)\left(\dfrac{305}{3}\right)\\
&=&\left(-\dfrac{34465}{12}\right)\sqrt{5}+\dfrac{56425}{108}\\
&=&\left(-\dfrac{34465}{12}\right)\sqrt{5}+\dfrac{56425}{108}\\
\end{eqnarray*}