L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{73}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{1}{3}\right)\sqrt{175}\right)\) et \( Y=\left(0\right)\sqrt{49}+\left(-\dfrac{11}{2}\right)\sqrt{175}+\left(-\dfrac{43}{3}\right)\sqrt{28}+\left(\left(4\right)\sqrt{49}\right)-\left(\left(\dfrac{47}{2}\right)\sqrt{63}\right)+\left(-2\right)\sqrt{28}+\left(\dfrac{76}{5}\right)\sqrt{63}+\left(\dfrac{20}{9}\right)\sqrt{28}\) . 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{73}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{1}{3}\right)\sqrt{175}\right)\right)+\left(\left(0\right)\sqrt{49}+\left(-\dfrac{11}{2}\right)\sqrt{175}+\left(-\dfrac{43}{3}\right)\sqrt{28}+\left(\left(4\right)\sqrt{49}\right)-\left(\left(\dfrac{47}{2}\right)\sqrt{63}\right)+\left(-2\right)\sqrt{28}+\left(\dfrac{76}{5}\right)\sqrt{63}+\left(\dfrac{20}{9}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{511}{4}-\left(\left(\dfrac{5}{3}\right)\sqrt{7}\right)\right)+\left(0+\left(-\dfrac{55}{2}\right)\sqrt{7}+\left(-\dfrac{86}{3}\right)\sqrt{7}+28-\left(\left(\dfrac{141}{2}\right)\sqrt{7}\right)+\left(-4\right)\sqrt{7}+\left(\dfrac{228}{5}\right)\sqrt{7}+\left(\dfrac{40}{9}\right)\sqrt{7}\right)\\
&=&-\dfrac{511}{4}-\left(\left(\dfrac{5}{3}\right)\sqrt{7}\right)+0+\left(-\dfrac{55}{2}\right)\sqrt{7}+\left(-\dfrac{86}{3}\right)\sqrt{7}+28-\left(\left(\dfrac{141}{2}\right)\sqrt{7}\right)+\left(-4\right)\sqrt{7}+\left(\dfrac{228}{5}\right)\sqrt{7}+\left(\dfrac{40}{9}\right)\sqrt{7}\\
&=&-\dfrac{399}{4}+\left(-\dfrac{3703}{45}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{73}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{1}{3}\right)\sqrt{175}\right)\right)-\left(\left(0\right)\sqrt{49}+\left(-\dfrac{11}{2}\right)\sqrt{175}+\left(-\dfrac{43}{3}\right)\sqrt{28}+\left(\left(4\right)\sqrt{49}\right)-\left(\left(\dfrac{47}{2}\right)\sqrt{63}\right)+\left(-2\right)\sqrt{28}+\left(\dfrac{76}{5}\right)\sqrt{63}+\left(\dfrac{20}{9}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{511}{4}-\left(\left(\dfrac{5}{3}\right)\sqrt{7}\right)\right)-\left(0+\left(-\dfrac{55}{2}\right)\sqrt{7}+\left(-\dfrac{86}{3}\right)\sqrt{7}+28-\left(\left(\dfrac{141}{2}\right)\sqrt{7}\right)+\left(-4\right)\sqrt{7}+\left(\dfrac{228}{5}\right)\sqrt{7}+\left(\dfrac{40}{9}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{511}{4}+\left(-\dfrac{5}{3}\right)\sqrt{7}\right)-\left(28+\left(-\dfrac{3628}{45}\right)\sqrt{7}\right)\\
&=&-\dfrac{511}{4}+\left(-\dfrac{5}{3}\right)\sqrt{7}+-28+\left(\dfrac{3628}{45}\right)\sqrt{7}\\
&=&-\dfrac{623}{4}+\left(\dfrac{3553}{45}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{73}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{1}{3}\right)\sqrt{175}\right)\right)\times\left(\left(0\right)\sqrt{49}+\left(-\dfrac{11}{2}\right)\sqrt{175}+\left(-\dfrac{43}{3}\right)\sqrt{28}+\left(\left(4\right)\sqrt{49}\right)-\left(\left(\dfrac{47}{2}\right)\sqrt{63}\right)+\left(-2\right)\sqrt{28}+\left(\dfrac{76}{5}\right)\sqrt{63}+\left(\dfrac{20}{9}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{511}{4}-\left(\left(\dfrac{5}{3}\right)\sqrt{7}\right)\right)\times\left(0+\left(-\dfrac{55}{2}\right)\sqrt{7}+\left(-\dfrac{86}{3}\right)\sqrt{7}+28-\left(\left(\dfrac{141}{2}\right)\sqrt{7}\right)+\left(-4\right)\sqrt{7}+\left(\dfrac{228}{5}\right)\sqrt{7}+\left(\dfrac{40}{9}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{511}{4}+\left(-\dfrac{5}{3}\right)\sqrt{7}\right)\left(28+\left(-\dfrac{3628}{45}\right)\sqrt{7}\right)\\
&=&-3577+\left(\dfrac{461377}{45}\right)\sqrt{7}+\left(\dfrac{3628}{27}\right)\sqrt{49}\\
&=&-\dfrac{71183}{27}+\left(\dfrac{461377}{45}\right)\sqrt{7}\\
\end{eqnarray*}