L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{41}{6}\right)\sqrt{25}\) et \( Y=\left(-1-\dfrac{38}{9}+\left(-3\right)\sqrt{45}+\left(\dfrac{73}{6}\right)\sqrt{125}\right)-\left(1-\left(\left(\dfrac{22}{3}\right)\sqrt{25}\right)-\dfrac{72}{5}-\left(\left(\dfrac{59}{5}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{79}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{4}\right)\sqrt{125}\right)\right)-\left(\dfrac{21}{5}+\left(-\dfrac{49}{6}\right)\sqrt{25}+\left(\dfrac{5}{8}\right)\sqrt{20}\right)-\left(\left(\dfrac{73}{6}\right)\sqrt{125}+\left(-\dfrac{59}{2}\right)\sqrt{20}+\left(-3\right)\sqrt{45}+\left(-\dfrac{18}{7}\right)\sqrt{45}+\left(0\right)\sqrt{125}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{41}{6}\right)\sqrt{25}\right)+\left(\left(-1-\dfrac{38}{9}+\left(-3\right)\sqrt{45}+\left(\dfrac{73}{6}\right)\sqrt{125}\right)-\left(1-\left(\left(\dfrac{22}{3}\right)\sqrt{25}\right)-\dfrac{72}{5}-\left(\left(\dfrac{59}{5}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{79}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{4}\right)\sqrt{125}\right)\right)-\left(\dfrac{21}{5}+\left(-\dfrac{49}{6}\right)\sqrt{25}+\left(\dfrac{5}{8}\right)\sqrt{20}\right)-\left(\left(\dfrac{73}{6}\right)\sqrt{125}+\left(-\dfrac{59}{2}\right)\sqrt{20}+\left(-3\right)\sqrt{45}+\left(-\dfrac{18}{7}\right)\sqrt{45}+\left(0\right)\sqrt{125}\right)\right)\\
&=&\left(\dfrac{205}{6}\right)+\left(\left(-1-\dfrac{38}{9}+\left(-9\right)\sqrt{5}+\left(\dfrac{365}{6}\right)\sqrt{5}\right)-\left(1-\dfrac{110}{3}-\dfrac{72}{5}-\left(\left(59\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{237}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{85}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{185}{4}\right)\sqrt{5}\right)\right)-\left(\dfrac{21}{5}-\dfrac{245}{6}+\left(\dfrac{5}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{365}{6}\right)\sqrt{5}+\left(-59\right)\sqrt{5}+\left(-9\right)\sqrt{5}+\left(-\dfrac{54}{7}\right)\sqrt{5}+\left(0\right)\sqrt{5}\right)\right)\\
&=&\dfrac{205}{6}+\left(-1-\dfrac{38}{9}+\left(-9\right)\sqrt{5}+\left(\dfrac{365}{6}\right)\sqrt{5}\right)-\left(1-\dfrac{110}{3}-\dfrac{72}{5}-\left(\left(59\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{237}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{85}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{185}{4}\right)\sqrt{5}\right)\right)-\left(\dfrac{21}{5}-\dfrac{245}{6}+\left(\dfrac{5}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{365}{6}\right)\sqrt{5}+\left(-59\right)\sqrt{5}+\left(-9\right)\sqrt{5}+\left(-\dfrac{54}{7}\right)\sqrt{5}+\left(0\right)\sqrt{5}\right)\\
&=&\dfrac{5204}{45}+\left(\dfrac{5249}{28}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{41}{6}\right)\sqrt{25}\right)-\left(\left(-1-\dfrac{38}{9}+\left(-3\right)\sqrt{45}+\left(\dfrac{73}{6}\right)\sqrt{125}\right)-\left(1-\left(\left(\dfrac{22}{3}\right)\sqrt{25}\right)-\dfrac{72}{5}-\left(\left(\dfrac{59}{5}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{79}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{4}\right)\sqrt{125}\right)\right)-\left(\dfrac{21}{5}+\left(-\dfrac{49}{6}\right)\sqrt{25}+\left(\dfrac{5}{8}\right)\sqrt{20}\right)-\left(\left(\dfrac{73}{6}\right)\sqrt{125}+\left(-\dfrac{59}{2}\right)\sqrt{20}+\left(-3\right)\sqrt{45}+\left(-\dfrac{18}{7}\right)\sqrt{45}+\left(0\right)\sqrt{125}\right)\right)\\
&=&\left(\dfrac{205}{6}\right)-\left(\left(-1-\dfrac{38}{9}+\left(-9\right)\sqrt{5}+\left(\dfrac{365}{6}\right)\sqrt{5}\right)-\left(1-\dfrac{110}{3}-\dfrac{72}{5}-\left(\left(59\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{237}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{85}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{185}{4}\right)\sqrt{5}\right)\right)-\left(\dfrac{21}{5}-\dfrac{245}{6}+\left(\dfrac{5}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{365}{6}\right)\sqrt{5}+\left(-59\right)\sqrt{5}+\left(-9\right)\sqrt{5}+\left(-\dfrac{54}{7}\right)\sqrt{5}+\left(0\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{205}{6}\right)-\left(\dfrac{7333}{90}+\left(\dfrac{5249}{28}\right)\sqrt{5}\right)\\
&=&\dfrac{205}{6}+-\dfrac{7333}{90}+\left(-\dfrac{5249}{28}\right)\sqrt{5}\\
&=&-\dfrac{2129}{45}+\left(-\dfrac{5249}{28}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{41}{6}\right)\sqrt{25}\right)\times\left(\left(-1-\dfrac{38}{9}+\left(-3\right)\sqrt{45}+\left(\dfrac{73}{6}\right)\sqrt{125}\right)-\left(1-\left(\left(\dfrac{22}{3}\right)\sqrt{25}\right)-\dfrac{72}{5}-\left(\left(\dfrac{59}{5}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{79}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{4}\right)\sqrt{125}\right)\right)-\left(\dfrac{21}{5}+\left(-\dfrac{49}{6}\right)\sqrt{25}+\left(\dfrac{5}{8}\right)\sqrt{20}\right)-\left(\left(\dfrac{73}{6}\right)\sqrt{125}+\left(-\dfrac{59}{2}\right)\sqrt{20}+\left(-3\right)\sqrt{45}+\left(-\dfrac{18}{7}\right)\sqrt{45}+\left(0\right)\sqrt{125}\right)\right)\\
&=&\left(\dfrac{205}{6}\right)\times\left(\left(-1-\dfrac{38}{9}+\left(-9\right)\sqrt{5}+\left(\dfrac{365}{6}\right)\sqrt{5}\right)-\left(1-\dfrac{110}{3}-\dfrac{72}{5}-\left(\left(59\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{237}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{85}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{185}{4}\right)\sqrt{5}\right)\right)-\left(\dfrac{21}{5}-\dfrac{245}{6}+\left(\dfrac{5}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{365}{6}\right)\sqrt{5}+\left(-59\right)\sqrt{5}+\left(-9\right)\sqrt{5}+\left(-\dfrac{54}{7}\right)\sqrt{5}+\left(0\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{205}{6}\right)\left(\dfrac{7333}{90}+\left(\dfrac{5249}{28}\right)\sqrt{5}\right)\\
&=&\dfrac{300653}{108}+\left(\dfrac{1076045}{168}\right)\sqrt{5}\\
&=&\dfrac{300653}{108}+\left(\dfrac{1076045}{168}\right)\sqrt{5}\\
\end{eqnarray*}