L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-6\right)\sqrt{175}+\left(\left(\dfrac{5}{9}\right)\sqrt{28}\right)-\left(\left(-\dfrac{41}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{63}\right)-\left(\left(\dfrac{68}{9}\right)\sqrt{49}\right)\) et \( Y=\dfrac{29}{7}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-6\right)\sqrt{175}+\left(\left(\dfrac{5}{9}\right)\sqrt{28}\right)-\left(\left(-\dfrac{41}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{63}\right)-\left(\left(\dfrac{68}{9}\right)\sqrt{49}\right)\right)+\left(\dfrac{29}{7}\right)\\
&=&\left(\left(-30\right)\sqrt{7}+\left(\left(\dfrac{10}{9}\right)\sqrt{7}\right)-\left(\left(-\dfrac{41}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{162}{7}\right)\sqrt{7}\right)-\dfrac{476}{9}\right)+\left(\dfrac{29}{7}\right)\\
&=&\left(-30\right)\sqrt{7}+\left(\left(\dfrac{10}{9}\right)\sqrt{7}\right)-\left(\left(-\dfrac{41}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{162}{7}\right)\sqrt{7}\right)-\dfrac{476}{9}+\dfrac{29}{7}\\
&=&\left(-\dfrac{2417}{63}\right)\sqrt{7}-\dfrac{3071}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-6\right)\sqrt{175}+\left(\left(\dfrac{5}{9}\right)\sqrt{28}\right)-\left(\left(-\dfrac{41}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{63}\right)-\left(\left(\dfrac{68}{9}\right)\sqrt{49}\right)\right)-\left(\dfrac{29}{7}\right)\\
&=&\left(\left(-30\right)\sqrt{7}+\left(\left(\dfrac{10}{9}\right)\sqrt{7}\right)-\left(\left(-\dfrac{41}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{162}{7}\right)\sqrt{7}\right)-\dfrac{476}{9}\right)-\left(\dfrac{29}{7}\right)\\
&=&\left(\left(-\dfrac{2417}{63}\right)\sqrt{7}-\dfrac{476}{9}\right)-\left(\dfrac{29}{7}\right)\\
&=&\left(-\dfrac{2417}{63}\right)\sqrt{7}-\dfrac{476}{9}+-\dfrac{29}{7}\\
&=&\left(-\dfrac{2417}{63}\right)\sqrt{7}-\dfrac{3593}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-6\right)\sqrt{175}+\left(\left(\dfrac{5}{9}\right)\sqrt{28}\right)-\left(\left(-\dfrac{41}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{63}\right)-\left(\left(\dfrac{68}{9}\right)\sqrt{49}\right)\right)\times\left(\dfrac{29}{7}\right)\\
&=&\left(\left(-30\right)\sqrt{7}+\left(\left(\dfrac{10}{9}\right)\sqrt{7}\right)-\left(\left(-\dfrac{41}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{162}{7}\right)\sqrt{7}\right)-\dfrac{476}{9}\right)\times\left(\dfrac{29}{7}\right)\\
&=&\left(\left(-\dfrac{2417}{63}\right)\sqrt{7}-\dfrac{476}{9}\right)\left(\dfrac{29}{7}\right)\\
&=&\left(-\dfrac{70093}{441}\right)\sqrt{7}-\dfrac{1972}{9}\\
&=&\left(-\dfrac{70093}{441}\right)\sqrt{7}-\dfrac{1972}{9}\\
\end{eqnarray*}