L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{34}{3}\right)\sqrt{28}-5+\left(\dfrac{11}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{39}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{62}{3}\right)\sqrt{175}+\left(\dfrac{34}{3}\right)\sqrt{28}+\left(\dfrac{57}{4}\right)\sqrt{28}\right)-\left(\left(\dfrac{59}{7}\right)\sqrt{28}-5+\left(-\dfrac{13}{4}\right)\sqrt{63}\right)\) et \( Y=\left(-9\right)\sqrt{49}-\dfrac{38}{9}+5+\left(-\dfrac{39}{5}\right)\sqrt{28}+\left(\left(-3\right)\sqrt{63}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{49}\right)+\dfrac{1}{2}+5\) . 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(\dfrac{34}{3}\right)\sqrt{28}-5+\left(\dfrac{11}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{39}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{62}{3}\right)\sqrt{175}+\left(\dfrac{34}{3}\right)\sqrt{28}+\left(\dfrac{57}{4}\right)\sqrt{28}\right)-\left(\left(\dfrac{59}{7}\right)\sqrt{28}-5+\left(-\dfrac{13}{4}\right)\sqrt{63}\right)\right)+\left(\left(-9\right)\sqrt{49}-\dfrac{38}{9}+5+\left(-\dfrac{39}{5}\right)\sqrt{28}+\left(\left(-3\right)\sqrt{63}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{49}\right)+\dfrac{1}{2}+5\right)\\
&=&\left(\left(\left(\dfrac{68}{3}\right)\sqrt{7}-5+\dfrac{77}{3}\right)-\left(\left(\dfrac{195}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{310}{3}\right)\sqrt{7}+\left(\dfrac{68}{3}\right)\sqrt{7}+\left(\dfrac{57}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{118}{7}\right)\sqrt{7}-5+\left(-\dfrac{39}{4}\right)\sqrt{7}\right)\right)+\left(-63-\dfrac{38}{9}+5+\left(-\dfrac{78}{5}\right)\sqrt{7}+\left(\left(-9\right)\sqrt{7}\right)+\dfrac{133}{2}+\dfrac{1}{2}+5\right)\\
&=&\left(\left(\dfrac{68}{3}\right)\sqrt{7}-5+\dfrac{77}{3}\right)-\left(\left(\dfrac{195}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{310}{3}\right)\sqrt{7}+\left(\dfrac{68}{3}\right)\sqrt{7}+\left(\dfrac{57}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{118}{7}\right)\sqrt{7}-5+\left(-\dfrac{39}{4}\right)\sqrt{7}\right)-63-\dfrac{38}{9}+5+\left(-\dfrac{78}{5}\right)\sqrt{7}+\left(\left(-9\right)\sqrt{7}\right)+\dfrac{133}{2}+\dfrac{1}{2}+5\\
&=&\left(-\dfrac{44581}{210}\right)\sqrt{7}+\dfrac{319}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{34}{3}\right)\sqrt{28}-5+\left(\dfrac{11}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{39}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{62}{3}\right)\sqrt{175}+\left(\dfrac{34}{3}\right)\sqrt{28}+\left(\dfrac{57}{4}\right)\sqrt{28}\right)-\left(\left(\dfrac{59}{7}\right)\sqrt{28}-5+\left(-\dfrac{13}{4}\right)\sqrt{63}\right)\right)-\left(\left(-9\right)\sqrt{49}-\dfrac{38}{9}+5+\left(-\dfrac{39}{5}\right)\sqrt{28}+\left(\left(-3\right)\sqrt{63}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{49}\right)+\dfrac{1}{2}+5\right)\\
&=&\left(\left(\left(\dfrac{68}{3}\right)\sqrt{7}-5+\dfrac{77}{3}\right)-\left(\left(\dfrac{195}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{310}{3}\right)\sqrt{7}+\left(\dfrac{68}{3}\right)\sqrt{7}+\left(\dfrac{57}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{118}{7}\right)\sqrt{7}-5+\left(-\dfrac{39}{4}\right)\sqrt{7}\right)\right)-\left(-63-\dfrac{38}{9}+5+\left(-\dfrac{78}{5}\right)\sqrt{7}+\left(\left(-9\right)\sqrt{7}\right)+\dfrac{133}{2}+\dfrac{1}{2}+5\right)\\
&=&\left(\left(-\dfrac{7883}{42}\right)\sqrt{7}+\dfrac{77}{3}\right)-\left(\dfrac{88}{9}+\left(-\dfrac{123}{5}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{7883}{42}\right)\sqrt{7}+\dfrac{77}{3}+-\dfrac{88}{9}+\left(\dfrac{123}{5}\right)\sqrt{7}\\
&=&\left(-\dfrac{34249}{210}\right)\sqrt{7}+\dfrac{143}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{34}{3}\right)\sqrt{28}-5+\left(\dfrac{11}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{39}{4}\right)\sqrt{175}\right)-\left(\left(\dfrac{62}{3}\right)\sqrt{175}+\left(\dfrac{34}{3}\right)\sqrt{28}+\left(\dfrac{57}{4}\right)\sqrt{28}\right)-\left(\left(\dfrac{59}{7}\right)\sqrt{28}-5+\left(-\dfrac{13}{4}\right)\sqrt{63}\right)\right)\times\left(\left(-9\right)\sqrt{49}-\dfrac{38}{9}+5+\left(-\dfrac{39}{5}\right)\sqrt{28}+\left(\left(-3\right)\sqrt{63}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{49}\right)+\dfrac{1}{2}+5\right)\\
&=&\left(\left(\left(\dfrac{68}{3}\right)\sqrt{7}-5+\dfrac{77}{3}\right)-\left(\left(\dfrac{195}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{310}{3}\right)\sqrt{7}+\left(\dfrac{68}{3}\right)\sqrt{7}+\left(\dfrac{57}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{118}{7}\right)\sqrt{7}-5+\left(-\dfrac{39}{4}\right)\sqrt{7}\right)\right)\times\left(-63-\dfrac{38}{9}+5+\left(-\dfrac{78}{5}\right)\sqrt{7}+\left(\left(-9\right)\sqrt{7}\right)+\dfrac{133}{2}+\dfrac{1}{2}+5\right)\\
&=&\left(\left(-\dfrac{7883}{42}\right)\sqrt{7}+\dfrac{77}{3}\right)\left(\dfrac{88}{9}+\left(-\dfrac{123}{5}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{2330933}{945}\right)\sqrt{7}+\left(\dfrac{323203}{70}\right)\sqrt{49}+\dfrac{6776}{27}\\
&=&\left(-\dfrac{2330933}{945}\right)\sqrt{7}+\dfrac{8794241}{270}\\
\end{eqnarray*}