L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{15}{7}-\dfrac{62}{3}+\dfrac{10}{7}-\left(\left(-\dfrac{69}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{4}{7}\right)\sqrt{75}\right)+\left(\left(-\dfrac{35}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{17}{4}\right)\sqrt{75}\right)+\dfrac{59}{3}-\dfrac{67}{2}\) et \( Y=\left(\left(\dfrac{77}{9}\right)\sqrt{75}+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(\dfrac{29}{4}\right)\sqrt{12}+\left(2\right)\sqrt{9}+0\right)-\left(-3-\left(\left(-\dfrac{27}{4}\right)\sqrt{12}\right)-\left(\left(-1\right)\sqrt{27}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{15}{7}-\dfrac{62}{3}+\dfrac{10}{7}-\left(\left(-\dfrac{69}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{4}{7}\right)\sqrt{75}\right)+\left(\left(-\dfrac{35}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{17}{4}\right)\sqrt{75}\right)+\dfrac{59}{3}-\dfrac{67}{2}\right)+\left(\left(\left(\dfrac{77}{9}\right)\sqrt{75}+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(\dfrac{29}{4}\right)\sqrt{12}+\left(2\right)\sqrt{9}+0\right)-\left(-3-\left(\left(-\dfrac{27}{4}\right)\sqrt{12}\right)-\left(\left(-1\right)\sqrt{27}\right)\right)\right)\\
&=&\left(-\dfrac{15}{7}-\dfrac{62}{3}+\dfrac{10}{7}-\left(\left(-\dfrac{138}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)+\left(\left(-\dfrac{175}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{85}{4}\right)\sqrt{3}\right)+\dfrac{59}{3}-\dfrac{67}{2}\right)+\left(\left(\left(\dfrac{385}{9}\right)\sqrt{3}-\dfrac{165}{7}+\left(\dfrac{29}{2}\right)\sqrt{3}+6+0\right)-\left(-3-\left(\left(-\dfrac{27}{2}\right)\sqrt{3}\right)-\left(\left(-3\right)\sqrt{3}\right)\right)\right)\\
&=&-\dfrac{15}{7}-\dfrac{62}{3}+\dfrac{10}{7}-\left(\left(-\dfrac{138}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)+\left(\left(-\dfrac{175}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{85}{4}\right)\sqrt{3}\right)+\dfrac{59}{3}-\dfrac{67}{2}+\left(\left(\dfrac{385}{9}\right)\sqrt{3}-\dfrac{165}{7}+\left(\dfrac{29}{2}\right)\sqrt{3}+6+0\right)-\left(-3-\left(\left(-\dfrac{27}{2}\right)\sqrt{3}\right)-\left(\left(-3\right)\sqrt{3}\right)\right)\\
&=&-\dfrac{697}{14}+\left(\dfrac{10193}{504}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{15}{7}-\dfrac{62}{3}+\dfrac{10}{7}-\left(\left(-\dfrac{69}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{4}{7}\right)\sqrt{75}\right)+\left(\left(-\dfrac{35}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{17}{4}\right)\sqrt{75}\right)+\dfrac{59}{3}-\dfrac{67}{2}\right)-\left(\left(\left(\dfrac{77}{9}\right)\sqrt{75}+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(\dfrac{29}{4}\right)\sqrt{12}+\left(2\right)\sqrt{9}+0\right)-\left(-3-\left(\left(-\dfrac{27}{4}\right)\sqrt{12}\right)-\left(\left(-1\right)\sqrt{27}\right)\right)\right)\\
&=&\left(-\dfrac{15}{7}-\dfrac{62}{3}+\dfrac{10}{7}-\left(\left(-\dfrac{138}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)+\left(\left(-\dfrac{175}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{85}{4}\right)\sqrt{3}\right)+\dfrac{59}{3}-\dfrac{67}{2}\right)-\left(\left(\left(\dfrac{385}{9}\right)\sqrt{3}-\dfrac{165}{7}+\left(\dfrac{29}{2}\right)\sqrt{3}+6+0\right)-\left(-3-\left(\left(-\dfrac{27}{2}\right)\sqrt{3}\right)-\left(\left(-3\right)\sqrt{3}\right)\right)\right)\\
&=&\left(-\dfrac{493}{14}+\left(-\dfrac{1151}{56}\right)\sqrt{3}\right)-\left(\left(\dfrac{367}{9}\right)\sqrt{3}-\dfrac{102}{7}\right)\\
&=&-\dfrac{493}{14}+\left(-\dfrac{1151}{56}\right)\sqrt{3}+\left(-\dfrac{367}{9}\right)\sqrt{3}+\dfrac{102}{7}\\
&=&-\dfrac{289}{14}+\left(-\dfrac{30911}{504}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{15}{7}-\dfrac{62}{3}+\dfrac{10}{7}-\left(\left(-\dfrac{69}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{4}{7}\right)\sqrt{75}\right)+\left(\left(-\dfrac{35}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{17}{4}\right)\sqrt{75}\right)+\dfrac{59}{3}-\dfrac{67}{2}\right)\times\left(\left(\left(\dfrac{77}{9}\right)\sqrt{75}+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(\dfrac{29}{4}\right)\sqrt{12}+\left(2\right)\sqrt{9}+0\right)-\left(-3-\left(\left(-\dfrac{27}{4}\right)\sqrt{12}\right)-\left(\left(-1\right)\sqrt{27}\right)\right)\right)\\
&=&\left(-\dfrac{15}{7}-\dfrac{62}{3}+\dfrac{10}{7}-\left(\left(-\dfrac{138}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{20}{7}\right)\sqrt{3}\right)+\left(\left(-\dfrac{175}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{85}{4}\right)\sqrt{3}\right)+\dfrac{59}{3}-\dfrac{67}{2}\right)\times\left(\left(\left(\dfrac{385}{9}\right)\sqrt{3}-\dfrac{165}{7}+\left(\dfrac{29}{2}\right)\sqrt{3}+6+0\right)-\left(-3-\left(\left(-\dfrac{27}{2}\right)\sqrt{3}\right)-\left(\left(-3\right)\sqrt{3}\right)\right)\right)\\
&=&\left(-\dfrac{493}{14}+\left(-\dfrac{1151}{56}\right)\sqrt{3}\right)\left(\left(\dfrac{367}{9}\right)\sqrt{3}-\dfrac{102}{7}\right)\\
&=&\left(-\dfrac{2004725}{1764}\right)\sqrt{3}+\dfrac{25143}{49}+\left(-\dfrac{422417}{504}\right)\sqrt{9}\\
&=&\left(-\dfrac{2004725}{1764}\right)\sqrt{3}-\dfrac{2353487}{1176}\\
\end{eqnarray*}