L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{49}{9}\right)\sqrt{28}+\left(-\dfrac{35}{3}\right)\sqrt{28}+\left(\dfrac{5}{2}\right)\sqrt{63}+\left(\dfrac{5}{2}\right)\sqrt{175}\) et \( Y=\left(7-\left(\left(-2\right)\sqrt{63}\right)-\left(\left(\dfrac{77}{8}\right)\sqrt{28}\right)-7\right)-\dfrac{31}{3}-\left(\left(\dfrac{35}{8}\right)\sqrt{63}+\left(-3\right)\sqrt{28}\right)\) . 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{49}{9}\right)\sqrt{28}+\left(-\dfrac{35}{3}\right)\sqrt{28}+\left(\dfrac{5}{2}\right)\sqrt{63}+\left(\dfrac{5}{2}\right)\sqrt{175}\right)+\left(\left(7-\left(\left(-2\right)\sqrt{63}\right)-\left(\left(\dfrac{77}{8}\right)\sqrt{28}\right)-7\right)-\dfrac{31}{3}-\left(\left(\dfrac{35}{8}\right)\sqrt{63}+\left(-3\right)\sqrt{28}\right)\right)\\
&=&\left(\left(-\dfrac{98}{9}\right)\sqrt{7}+\left(-\dfrac{70}{3}\right)\sqrt{7}+\left(\dfrac{15}{2}\right)\sqrt{7}+\left(\dfrac{25}{2}\right)\sqrt{7}\right)+\left(\left(7-\left(\left(-6\right)\sqrt{7}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{7}\right)-7\right)-\dfrac{31}{3}-\left(\left(\dfrac{105}{8}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{98}{9}\right)\sqrt{7}+\left(-\dfrac{70}{3}\right)\sqrt{7}+\left(\dfrac{15}{2}\right)\sqrt{7}+\left(\dfrac{25}{2}\right)\sqrt{7}+\left(7-\left(\left(-6\right)\sqrt{7}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{7}\right)-7\right)-\dfrac{31}{3}-\left(\left(\dfrac{105}{8}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{2491}{72}\right)\sqrt{7}-\dfrac{31}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{49}{9}\right)\sqrt{28}+\left(-\dfrac{35}{3}\right)\sqrt{28}+\left(\dfrac{5}{2}\right)\sqrt{63}+\left(\dfrac{5}{2}\right)\sqrt{175}\right)-\left(\left(7-\left(\left(-2\right)\sqrt{63}\right)-\left(\left(\dfrac{77}{8}\right)\sqrt{28}\right)-7\right)-\dfrac{31}{3}-\left(\left(\dfrac{35}{8}\right)\sqrt{63}+\left(-3\right)\sqrt{28}\right)\right)\\
&=&\left(\left(-\dfrac{98}{9}\right)\sqrt{7}+\left(-\dfrac{70}{3}\right)\sqrt{7}+\left(\dfrac{15}{2}\right)\sqrt{7}+\left(\dfrac{25}{2}\right)\sqrt{7}\right)-\left(\left(7-\left(\left(-6\right)\sqrt{7}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{7}\right)-7\right)-\dfrac{31}{3}-\left(\left(\dfrac{105}{8}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{128}{9}\right)\sqrt{7}\right)-\left(-\dfrac{31}{3}+\left(-\dfrac{163}{8}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{128}{9}\right)\sqrt{7}+\dfrac{31}{3}+\left(\dfrac{163}{8}\right)\sqrt{7}\\
&=&\left(\dfrac{443}{72}\right)\sqrt{7}+\dfrac{31}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{49}{9}\right)\sqrt{28}+\left(-\dfrac{35}{3}\right)\sqrt{28}+\left(\dfrac{5}{2}\right)\sqrt{63}+\left(\dfrac{5}{2}\right)\sqrt{175}\right)\times\left(\left(7-\left(\left(-2\right)\sqrt{63}\right)-\left(\left(\dfrac{77}{8}\right)\sqrt{28}\right)-7\right)-\dfrac{31}{3}-\left(\left(\dfrac{35}{8}\right)\sqrt{63}+\left(-3\right)\sqrt{28}\right)\right)\\
&=&\left(\left(-\dfrac{98}{9}\right)\sqrt{7}+\left(-\dfrac{70}{3}\right)\sqrt{7}+\left(\dfrac{15}{2}\right)\sqrt{7}+\left(\dfrac{25}{2}\right)\sqrt{7}\right)\times\left(\left(7-\left(\left(-6\right)\sqrt{7}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{7}\right)-7\right)-\dfrac{31}{3}-\left(\left(\dfrac{105}{8}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{128}{9}\right)\sqrt{7}\right)\left(-\dfrac{31}{3}+\left(-\dfrac{163}{8}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{3968}{27}\right)\sqrt{7}+\left(\dfrac{2608}{9}\right)\sqrt{49}\\
&=&\left(\dfrac{3968}{27}\right)\sqrt{7}+\dfrac{18256}{9}\\
\end{eqnarray*}