L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{7}{6}\right)\sqrt{12}+\left(-\dfrac{35}{4}\right)\sqrt{9}+\left(-7\right)\sqrt{27}+\left(0\right)\sqrt{12}+\left(-7\right)\sqrt{12}-\dfrac{32}{5}-\left(\left(-\dfrac{74}{9}\right)\sqrt{9}\right)-\left(\left(-9\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{27}\right)\) et \( Y=-\dfrac{67}{9}-\left(0+\dfrac{61}{2}+\left(-\dfrac{19}{3}\right)\sqrt{9}+\left(-\dfrac{11}{3}\right)\sqrt{75}\right)-\left(\left(-\dfrac{47}{2}\right)\sqrt{27}\right)-\left(\left(\left(\dfrac{79}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{5}{7}\right)\sqrt{12}\right)-\left(\left(-2\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{27}\right)\) . 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{7}{6}\right)\sqrt{12}+\left(-\dfrac{35}{4}\right)\sqrt{9}+\left(-7\right)\sqrt{27}+\left(0\right)\sqrt{12}+\left(-7\right)\sqrt{12}-\dfrac{32}{5}-\left(\left(-\dfrac{74}{9}\right)\sqrt{9}\right)-\left(\left(-9\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{27}\right)\right)+\left(-\dfrac{67}{9}-\left(0+\dfrac{61}{2}+\left(-\dfrac{19}{3}\right)\sqrt{9}+\left(-\dfrac{11}{3}\right)\sqrt{75}\right)-\left(\left(-\dfrac{47}{2}\right)\sqrt{27}\right)-\left(\left(\left(\dfrac{79}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{5}{7}\right)\sqrt{12}\right)-\left(\left(-2\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\dfrac{7}{3}\right)\sqrt{3}-\dfrac{105}{4}+\left(-21\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(-14\right)\sqrt{3}-\dfrac{32}{5}+\dfrac{74}{3}-\left(\left(-27\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)+\left(-\dfrac{67}{9}-\left(0+\dfrac{61}{2}-19+\left(-\dfrac{55}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{141}{2}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{395}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{7}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{7}{3}\right)\sqrt{3}-\dfrac{105}{4}+\left(-21\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(-14\right)\sqrt{3}-\dfrac{32}{5}+\dfrac{74}{3}-\left(\left(-27\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)-\dfrac{67}{9}-\left(0+\dfrac{61}{2}-19+\left(-\dfrac{55}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{141}{2}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{395}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{7}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{9193}{168}\right)\sqrt{3}-\dfrac{4847}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{7}{6}\right)\sqrt{12}+\left(-\dfrac{35}{4}\right)\sqrt{9}+\left(-7\right)\sqrt{27}+\left(0\right)\sqrt{12}+\left(-7\right)\sqrt{12}-\dfrac{32}{5}-\left(\left(-\dfrac{74}{9}\right)\sqrt{9}\right)-\left(\left(-9\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{27}\right)\right)-\left(-\dfrac{67}{9}-\left(0+\dfrac{61}{2}+\left(-\dfrac{19}{3}\right)\sqrt{9}+\left(-\dfrac{11}{3}\right)\sqrt{75}\right)-\left(\left(-\dfrac{47}{2}\right)\sqrt{27}\right)-\left(\left(\left(\dfrac{79}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{5}{7}\right)\sqrt{12}\right)-\left(\left(-2\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\dfrac{7}{3}\right)\sqrt{3}-\dfrac{105}{4}+\left(-21\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(-14\right)\sqrt{3}-\dfrac{32}{5}+\dfrac{74}{3}-\left(\left(-27\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)-\left(-\dfrac{67}{9}-\left(0+\dfrac{61}{2}-19+\left(-\dfrac{55}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{141}{2}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{395}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{7}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{17}{3}\right)\sqrt{3}-\dfrac{479}{60}\right)-\left(-\dfrac{341}{18}+\left(\dfrac{10145}{168}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{17}{3}\right)\sqrt{3}-\dfrac{479}{60}+\dfrac{341}{18}+\left(-\dfrac{10145}{168}\right)\sqrt{3}\\
&=&\left(-\dfrac{3699}{56}\right)\sqrt{3}+\dfrac{1973}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{7}{6}\right)\sqrt{12}+\left(-\dfrac{35}{4}\right)\sqrt{9}+\left(-7\right)\sqrt{27}+\left(0\right)\sqrt{12}+\left(-7\right)\sqrt{12}-\dfrac{32}{5}-\left(\left(-\dfrac{74}{9}\right)\sqrt{9}\right)-\left(\left(-9\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{27}\right)\right)\times\left(-\dfrac{67}{9}-\left(0+\dfrac{61}{2}+\left(-\dfrac{19}{3}\right)\sqrt{9}+\left(-\dfrac{11}{3}\right)\sqrt{75}\right)-\left(\left(-\dfrac{47}{2}\right)\sqrt{27}\right)-\left(\left(\left(\dfrac{79}{8}\right)\sqrt{75}\right)-\left(\left(\dfrac{5}{7}\right)\sqrt{12}\right)-\left(\left(-2\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\dfrac{7}{3}\right)\sqrt{3}-\dfrac{105}{4}+\left(-21\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(-14\right)\sqrt{3}-\dfrac{32}{5}+\dfrac{74}{3}-\left(\left(-27\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)\times\left(-\dfrac{67}{9}-\left(0+\dfrac{61}{2}-19+\left(-\dfrac{55}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{141}{2}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{395}{8}\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{7}\right)\sqrt{3}\right)-\left(\left(-6\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{17}{3}\right)\sqrt{3}-\dfrac{479}{60}\right)\left(-\dfrac{341}{18}+\left(\dfrac{10145}{168}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{2266409}{6048}\right)\sqrt{3}+\left(-\dfrac{172465}{504}\right)\sqrt{9}+\dfrac{163339}{1080}\\
&=&\left(-\dfrac{2266409}{6048}\right)\sqrt{3}-\dfrac{827194}{945}\\
\end{eqnarray*}