L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\dfrac{47}{7}-\left(-7+7-\left(\left(\dfrac{37}{8}\right)\sqrt{20}\right)\right)-\left(-7+\left(-\dfrac{35}{2}\right)\sqrt{125}-\dfrac{32}{7}\right)\) et \( Y=0-\left(\left(\dfrac{34}{3}\right)\sqrt{20}\right)-\left(\left(-7\right)\sqrt{125}\right)-\dfrac{78}{7}+\left(\left(-\dfrac{39}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{45}\right)+\left(\dfrac{1}{4}\right)\sqrt{125}+\left(\dfrac{21}{8}\right)\sqrt{25}+\dfrac{49}{6}+\dfrac{73}{9}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{47}{7}-\left(-7+7-\left(\left(\dfrac{37}{8}\right)\sqrt{20}\right)\right)-\left(-7+\left(-\dfrac{35}{2}\right)\sqrt{125}-\dfrac{32}{7}\right)\right)+\left(0-\left(\left(\dfrac{34}{3}\right)\sqrt{20}\right)-\left(\left(-7\right)\sqrt{125}\right)-\dfrac{78}{7}+\left(\left(-\dfrac{39}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{45}\right)+\left(\dfrac{1}{4}\right)\sqrt{125}+\left(\dfrac{21}{8}\right)\sqrt{25}+\dfrac{49}{6}+\dfrac{73}{9}\right)\\
&=&\left(\dfrac{47}{7}-\left(-7+7-\left(\left(\dfrac{37}{4}\right)\sqrt{5}\right)\right)-\left(-7+\left(-\dfrac{175}{2}\right)\sqrt{5}-\dfrac{32}{7}\right)\right)+\left(0-\left(\left(\dfrac{68}{3}\right)\sqrt{5}\right)-\left(\left(-35\right)\sqrt{5}\right)-\dfrac{78}{7}+\left(\left(-\dfrac{195}{4}\right)\sqrt{5}\right)-\left(\left(7\right)\sqrt{5}\right)+\left(\dfrac{5}{4}\right)\sqrt{5}+\dfrac{105}{8}+\dfrac{49}{6}+\dfrac{73}{9}\right)\\
&=&\dfrac{47}{7}-\left(-7+7-\left(\left(\dfrac{37}{4}\right)\sqrt{5}\right)\right)-\left(-7+\left(-\dfrac{175}{2}\right)\sqrt{5}-\dfrac{32}{7}\right)+0-\left(\left(\dfrac{68}{3}\right)\sqrt{5}\right)-\left(\left(-35\right)\sqrt{5}\right)-\dfrac{78}{7}+\left(\left(-\dfrac{195}{4}\right)\sqrt{5}\right)-\left(\left(7\right)\sqrt{5}\right)+\left(\dfrac{5}{4}\right)\sqrt{5}+\dfrac{105}{8}+\dfrac{49}{6}+\dfrac{73}{9}\\
&=&\dfrac{18419}{504}+\left(\dfrac{655}{12}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{47}{7}-\left(-7+7-\left(\left(\dfrac{37}{8}\right)\sqrt{20}\right)\right)-\left(-7+\left(-\dfrac{35}{2}\right)\sqrt{125}-\dfrac{32}{7}\right)\right)-\left(0-\left(\left(\dfrac{34}{3}\right)\sqrt{20}\right)-\left(\left(-7\right)\sqrt{125}\right)-\dfrac{78}{7}+\left(\left(-\dfrac{39}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{45}\right)+\left(\dfrac{1}{4}\right)\sqrt{125}+\left(\dfrac{21}{8}\right)\sqrt{25}+\dfrac{49}{6}+\dfrac{73}{9}\right)\\
&=&\left(\dfrac{47}{7}-\left(-7+7-\left(\left(\dfrac{37}{4}\right)\sqrt{5}\right)\right)-\left(-7+\left(-\dfrac{175}{2}\right)\sqrt{5}-\dfrac{32}{7}\right)\right)-\left(0-\left(\left(\dfrac{68}{3}\right)\sqrt{5}\right)-\left(\left(-35\right)\sqrt{5}\right)-\dfrac{78}{7}+\left(\left(-\dfrac{195}{4}\right)\sqrt{5}\right)-\left(\left(7\right)\sqrt{5}\right)+\left(\dfrac{5}{4}\right)\sqrt{5}+\dfrac{105}{8}+\dfrac{49}{6}+\dfrac{73}{9}\right)\\
&=&\left(\dfrac{128}{7}+\left(\dfrac{387}{4}\right)\sqrt{5}\right)-\left(\dfrac{9203}{504}+\left(-\dfrac{253}{6}\right)\sqrt{5}\right)\\
&=&\dfrac{128}{7}+\left(\dfrac{387}{4}\right)\sqrt{5}+-\dfrac{9203}{504}+\left(\dfrac{253}{6}\right)\sqrt{5}\\
&=&\dfrac{13}{504}+\left(\dfrac{1667}{12}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{47}{7}-\left(-7+7-\left(\left(\dfrac{37}{8}\right)\sqrt{20}\right)\right)-\left(-7+\left(-\dfrac{35}{2}\right)\sqrt{125}-\dfrac{32}{7}\right)\right)\times\left(0-\left(\left(\dfrac{34}{3}\right)\sqrt{20}\right)-\left(\left(-7\right)\sqrt{125}\right)-\dfrac{78}{7}+\left(\left(-\dfrac{39}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{45}\right)+\left(\dfrac{1}{4}\right)\sqrt{125}+\left(\dfrac{21}{8}\right)\sqrt{25}+\dfrac{49}{6}+\dfrac{73}{9}\right)\\
&=&\left(\dfrac{47}{7}-\left(-7+7-\left(\left(\dfrac{37}{4}\right)\sqrt{5}\right)\right)-\left(-7+\left(-\dfrac{175}{2}\right)\sqrt{5}-\dfrac{32}{7}\right)\right)\times\left(0-\left(\left(\dfrac{68}{3}\right)\sqrt{5}\right)-\left(\left(-35\right)\sqrt{5}\right)-\dfrac{78}{7}+\left(\left(-\dfrac{195}{4}\right)\sqrt{5}\right)-\left(\left(7\right)\sqrt{5}\right)+\left(\dfrac{5}{4}\right)\sqrt{5}+\dfrac{105}{8}+\dfrac{49}{6}+\dfrac{73}{9}\right)\\
&=&\left(\dfrac{128}{7}+\left(\dfrac{387}{4}\right)\sqrt{5}\right)\left(\dfrac{9203}{504}+\left(-\dfrac{253}{6}\right)\sqrt{5}\right)\\
&=&\dfrac{147248}{441}+\left(\dfrac{669043}{672}\right)\sqrt{5}+\left(-\dfrac{32637}{8}\right)\sqrt{25}\\
&=&-\dfrac{70786601}{3528}+\left(\dfrac{669043}{672}\right)\sqrt{5}\\
\end{eqnarray*}