L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{74}{9}\right)\sqrt{63}+\left(0\right)\sqrt{63}+\left(-\dfrac{34}{5}\right)\sqrt{28}+\left(-7\right)\sqrt{175}+\dfrac{5}{4}\) et \( Y=\left(\left(5\right)\sqrt{28}+\left(0\right)\sqrt{49}-\dfrac{39}{2}+\left(-\dfrac{26}{3}\right)\sqrt{49}+\left(\dfrac{28}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{19}{3}\right)\sqrt{28}\right)+\dfrac{13}{2}-\dfrac{61}{8}\) . 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{74}{9}\right)\sqrt{63}+\left(0\right)\sqrt{63}+\left(-\dfrac{34}{5}\right)\sqrt{28}+\left(-7\right)\sqrt{175}+\dfrac{5}{4}\right)+\left(\left(\left(5\right)\sqrt{28}+\left(0\right)\sqrt{49}-\dfrac{39}{2}+\left(-\dfrac{26}{3}\right)\sqrt{49}+\left(\dfrac{28}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{19}{3}\right)\sqrt{28}\right)+\dfrac{13}{2}-\dfrac{61}{8}\right)\\
&=&\left(\left(-\dfrac{74}{3}\right)\sqrt{7}+\left(0\right)\sqrt{7}+\left(-\dfrac{68}{5}\right)\sqrt{7}+\left(-35\right)\sqrt{7}+\dfrac{5}{4}\right)+\left(\left(\left(10\right)\sqrt{7}+0-\dfrac{39}{2}-\dfrac{182}{3}+\dfrac{196}{9}\right)-\left(\left(-\dfrac{38}{3}\right)\sqrt{7}\right)+\dfrac{13}{2}-\dfrac{61}{8}\right)\\
&=&\left(-\dfrac{74}{3}\right)\sqrt{7}+\left(0\right)\sqrt{7}+\left(-\dfrac{68}{5}\right)\sqrt{7}+\left(-35\right)\sqrt{7}+\dfrac{5}{4}+\left(\left(10\right)\sqrt{7}+0-\dfrac{39}{2}-\dfrac{182}{3}+\dfrac{196}{9}\right)-\left(\left(-\dfrac{38}{3}\right)\sqrt{7}\right)+\dfrac{13}{2}-\dfrac{61}{8}\\
&=&\left(-\dfrac{253}{5}\right)\sqrt{7}-\dfrac{4195}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{74}{9}\right)\sqrt{63}+\left(0\right)\sqrt{63}+\left(-\dfrac{34}{5}\right)\sqrt{28}+\left(-7\right)\sqrt{175}+\dfrac{5}{4}\right)-\left(\left(\left(5\right)\sqrt{28}+\left(0\right)\sqrt{49}-\dfrac{39}{2}+\left(-\dfrac{26}{3}\right)\sqrt{49}+\left(\dfrac{28}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{19}{3}\right)\sqrt{28}\right)+\dfrac{13}{2}-\dfrac{61}{8}\right)\\
&=&\left(\left(-\dfrac{74}{3}\right)\sqrt{7}+\left(0\right)\sqrt{7}+\left(-\dfrac{68}{5}\right)\sqrt{7}+\left(-35\right)\sqrt{7}+\dfrac{5}{4}\right)-\left(\left(\left(10\right)\sqrt{7}+0-\dfrac{39}{2}-\dfrac{182}{3}+\dfrac{196}{9}\right)-\left(\left(-\dfrac{38}{3}\right)\sqrt{7}\right)+\dfrac{13}{2}-\dfrac{61}{8}\right)\\
&=&\left(\left(-\dfrac{1099}{15}\right)\sqrt{7}+\dfrac{5}{4}\right)-\left(\left(\dfrac{68}{3}\right)\sqrt{7}-\dfrac{4285}{72}\right)\\
&=&\left(-\dfrac{1099}{15}\right)\sqrt{7}+\dfrac{5}{4}+\left(-\dfrac{68}{3}\right)\sqrt{7}+\dfrac{4285}{72}\\
&=&\left(-\dfrac{1439}{15}\right)\sqrt{7}+\dfrac{4375}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{74}{9}\right)\sqrt{63}+\left(0\right)\sqrt{63}+\left(-\dfrac{34}{5}\right)\sqrt{28}+\left(-7\right)\sqrt{175}+\dfrac{5}{4}\right)\times\left(\left(\left(5\right)\sqrt{28}+\left(0\right)\sqrt{49}-\dfrac{39}{2}+\left(-\dfrac{26}{3}\right)\sqrt{49}+\left(\dfrac{28}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{19}{3}\right)\sqrt{28}\right)+\dfrac{13}{2}-\dfrac{61}{8}\right)\\
&=&\left(\left(-\dfrac{74}{3}\right)\sqrt{7}+\left(0\right)\sqrt{7}+\left(-\dfrac{68}{5}\right)\sqrt{7}+\left(-35\right)\sqrt{7}+\dfrac{5}{4}\right)\times\left(\left(\left(10\right)\sqrt{7}+0-\dfrac{39}{2}-\dfrac{182}{3}+\dfrac{196}{9}\right)-\left(\left(-\dfrac{38}{3}\right)\sqrt{7}\right)+\dfrac{13}{2}-\dfrac{61}{8}\right)\\
&=&\left(\left(-\dfrac{1099}{15}\right)\sqrt{7}+\dfrac{5}{4}\right)\left(\left(\dfrac{68}{3}\right)\sqrt{7}-\dfrac{4285}{72}\right)\\
&=&\left(-\dfrac{74732}{45}\right)\sqrt{49}+\left(\dfrac{947963}{216}\right)\sqrt{7}-\dfrac{21425}{288}\\
&=&-\dfrac{16847093}{1440}+\left(\dfrac{947963}{216}\right)\sqrt{7}\\
\end{eqnarray*}