L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{73}{6}\right)\sqrt{175}\right)-\left(\left(-\dfrac{34}{3}\right)\sqrt{63}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{63}\right)\right)-\left(\left(-9\right)\sqrt{49}+\left(-\dfrac{79}{5}\right)\sqrt{63}+\left(-\dfrac{19}{9}\right)\sqrt{63}\right)-\left(\left(4\right)\sqrt{63}+\left(\dfrac{23}{3}\right)\sqrt{49}+\left(-\dfrac{34}{3}\right)\sqrt{63}+0\right)\) et \( Y=\left(\dfrac{35}{3}\right)\sqrt{175}+\left(-\dfrac{20}{9}\right)\sqrt{49}+\left(\dfrac{42}{5}\right)\sqrt{175}\) . 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(\left(\dfrac{73}{6}\right)\sqrt{175}\right)-\left(\left(-\dfrac{34}{3}\right)\sqrt{63}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{63}\right)\right)-\left(\left(-9\right)\sqrt{49}+\left(-\dfrac{79}{5}\right)\sqrt{63}+\left(-\dfrac{19}{9}\right)\sqrt{63}\right)-\left(\left(4\right)\sqrt{63}+\left(\dfrac{23}{3}\right)\sqrt{49}+\left(-\dfrac{34}{3}\right)\sqrt{63}+0\right)\right)+\left(\left(\dfrac{35}{3}\right)\sqrt{175}+\left(-\dfrac{20}{9}\right)\sqrt{49}+\left(\dfrac{42}{5}\right)\sqrt{175}\right)\\
&=&\left(\left(\left(\left(\dfrac{365}{6}\right)\sqrt{7}\right)-\left(\left(-34\right)\sqrt{7}\right)-\dfrac{161}{3}-\left(\left(\dfrac{57}{2}\right)\sqrt{7}\right)\right)-\left(-63+\left(-\dfrac{237}{5}\right)\sqrt{7}+\left(-\dfrac{19}{3}\right)\sqrt{7}\right)-\left(\left(12\right)\sqrt{7}+\dfrac{161}{3}+\left(-34\right)\sqrt{7}+0\right)\right)+\left(\left(\dfrac{175}{3}\right)\sqrt{7}-\dfrac{140}{9}+\left(42\right)\sqrt{7}\right)\\
&=&\left(\left(\left(\dfrac{365}{6}\right)\sqrt{7}\right)-\left(\left(-34\right)\sqrt{7}\right)-\dfrac{161}{3}-\left(\left(\dfrac{57}{2}\right)\sqrt{7}\right)\right)-\left(-63+\left(-\dfrac{237}{5}\right)\sqrt{7}+\left(-\dfrac{19}{3}\right)\sqrt{7}\right)-\left(\left(12\right)\sqrt{7}+\dfrac{161}{3}+\left(-34\right)\sqrt{7}+0\right)+\left(\dfrac{175}{3}\right)\sqrt{7}-\dfrac{140}{9}+\left(42\right)\sqrt{7}\\
&=&\left(\dfrac{1212}{5}\right)\sqrt{7}-\dfrac{539}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{73}{6}\right)\sqrt{175}\right)-\left(\left(-\dfrac{34}{3}\right)\sqrt{63}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{63}\right)\right)-\left(\left(-9\right)\sqrt{49}+\left(-\dfrac{79}{5}\right)\sqrt{63}+\left(-\dfrac{19}{9}\right)\sqrt{63}\right)-\left(\left(4\right)\sqrt{63}+\left(\dfrac{23}{3}\right)\sqrt{49}+\left(-\dfrac{34}{3}\right)\sqrt{63}+0\right)\right)-\left(\left(\dfrac{35}{3}\right)\sqrt{175}+\left(-\dfrac{20}{9}\right)\sqrt{49}+\left(\dfrac{42}{5}\right)\sqrt{175}\right)\\
&=&\left(\left(\left(\left(\dfrac{365}{6}\right)\sqrt{7}\right)-\left(\left(-34\right)\sqrt{7}\right)-\dfrac{161}{3}-\left(\left(\dfrac{57}{2}\right)\sqrt{7}\right)\right)-\left(-63+\left(-\dfrac{237}{5}\right)\sqrt{7}+\left(-\dfrac{19}{3}\right)\sqrt{7}\right)-\left(\left(12\right)\sqrt{7}+\dfrac{161}{3}+\left(-34\right)\sqrt{7}+0\right)\right)-\left(\left(\dfrac{175}{3}\right)\sqrt{7}-\dfrac{140}{9}+\left(42\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{2131}{15}\right)\sqrt{7}-\dfrac{133}{3}\right)-\left(\left(\dfrac{301}{3}\right)\sqrt{7}-\dfrac{140}{9}\right)\\
&=&\left(\dfrac{2131}{15}\right)\sqrt{7}-\dfrac{133}{3}+\left(-\dfrac{301}{3}\right)\sqrt{7}+\dfrac{140}{9}\\
&=&\left(\dfrac{626}{15}\right)\sqrt{7}-\dfrac{259}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{73}{6}\right)\sqrt{175}\right)-\left(\left(-\dfrac{34}{3}\right)\sqrt{63}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{63}\right)\right)-\left(\left(-9\right)\sqrt{49}+\left(-\dfrac{79}{5}\right)\sqrt{63}+\left(-\dfrac{19}{9}\right)\sqrt{63}\right)-\left(\left(4\right)\sqrt{63}+\left(\dfrac{23}{3}\right)\sqrt{49}+\left(-\dfrac{34}{3}\right)\sqrt{63}+0\right)\right)\times\left(\left(\dfrac{35}{3}\right)\sqrt{175}+\left(-\dfrac{20}{9}\right)\sqrt{49}+\left(\dfrac{42}{5}\right)\sqrt{175}\right)\\
&=&\left(\left(\left(\left(\dfrac{365}{6}\right)\sqrt{7}\right)-\left(\left(-34\right)\sqrt{7}\right)-\dfrac{161}{3}-\left(\left(\dfrac{57}{2}\right)\sqrt{7}\right)\right)-\left(-63+\left(-\dfrac{237}{5}\right)\sqrt{7}+\left(-\dfrac{19}{3}\right)\sqrt{7}\right)-\left(\left(12\right)\sqrt{7}+\dfrac{161}{3}+\left(-34\right)\sqrt{7}+0\right)\right)\times\left(\left(\dfrac{175}{3}\right)\sqrt{7}-\dfrac{140}{9}+\left(42\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{2131}{15}\right)\sqrt{7}-\dfrac{133}{3}\right)\left(\left(\dfrac{301}{3}\right)\sqrt{7}-\dfrac{140}{9}\right)\\
&=&\left(\dfrac{641431}{45}\right)\sqrt{49}+\left(-\dfrac{179767}{27}\right)\sqrt{7}+\dfrac{18620}{27}\\
&=&\dfrac{13563151}{135}+\left(-\dfrac{179767}{27}\right)\sqrt{7}\\
\end{eqnarray*}