L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(1\right)\sqrt{75}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{9}\right)\right)-\left(\left(-2\right)\sqrt{12}-\dfrac{32}{3}+\left(\dfrac{68}{3}\right)\sqrt{27}+\dfrac{53}{3}\right)\) et \( Y=\left(\left(\left(-6\right)\sqrt{12}\right)-\left(\left(6\right)\sqrt{9}\right)-\dfrac{23}{3}-\left(\left(\dfrac{49}{8}\right)\sqrt{12}\right)\right)-\left(\left(-9\right)\sqrt{27}+\left(\dfrac{49}{6}\right)\sqrt{75}+\left(\dfrac{47}{9}\right)\sqrt{75}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(1\right)\sqrt{75}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{9}\right)\right)-\left(\left(-2\right)\sqrt{12}-\dfrac{32}{3}+\left(\dfrac{68}{3}\right)\sqrt{27}+\dfrac{53}{3}\right)\right)+\left(\left(\left(\left(-6\right)\sqrt{12}\right)-\left(\left(6\right)\sqrt{9}\right)-\dfrac{23}{3}-\left(\left(\dfrac{49}{8}\right)\sqrt{12}\right)\right)-\left(\left(-9\right)\sqrt{27}+\left(\dfrac{49}{6}\right)\sqrt{75}+\left(\dfrac{47}{9}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-\dfrac{111}{2}\right)\sqrt{3}\right)-4\right)-\left(\left(-4\right)\sqrt{3}-\dfrac{32}{3}+\left(68\right)\sqrt{3}+\dfrac{53}{3}\right)\right)+\left(\left(\left(\left(-12\right)\sqrt{3}\right)-18-\dfrac{23}{3}-\left(\left(\dfrac{49}{4}\right)\sqrt{3}\right)\right)-\left(\left(-27\right)\sqrt{3}+\left(\dfrac{245}{6}\right)\sqrt{3}+\left(\dfrac{235}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-\dfrac{111}{2}\right)\sqrt{3}\right)-4\right)-\left(\left(-4\right)\sqrt{3}-\dfrac{32}{3}+\left(68\right)\sqrt{3}+\dfrac{53}{3}\right)+\left(\left(\left(-12\right)\sqrt{3}\right)-18-\dfrac{23}{3}-\left(\left(\dfrac{49}{4}\right)\sqrt{3}\right)\right)-\left(\left(-27\right)\sqrt{3}+\left(\dfrac{245}{6}\right)\sqrt{3}+\left(\dfrac{235}{9}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{2437}{36}\right)\sqrt{3}-\dfrac{110}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(1\right)\sqrt{75}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{9}\right)\right)-\left(\left(-2\right)\sqrt{12}-\dfrac{32}{3}+\left(\dfrac{68}{3}\right)\sqrt{27}+\dfrac{53}{3}\right)\right)-\left(\left(\left(\left(-6\right)\sqrt{12}\right)-\left(\left(6\right)\sqrt{9}\right)-\dfrac{23}{3}-\left(\left(\dfrac{49}{8}\right)\sqrt{12}\right)\right)-\left(\left(-9\right)\sqrt{27}+\left(\dfrac{49}{6}\right)\sqrt{75}+\left(\dfrac{47}{9}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-\dfrac{111}{2}\right)\sqrt{3}\right)-4\right)-\left(\left(-4\right)\sqrt{3}-\dfrac{32}{3}+\left(68\right)\sqrt{3}+\dfrac{53}{3}\right)\right)-\left(\left(\left(\left(-12\right)\sqrt{3}\right)-18-\dfrac{23}{3}-\left(\left(\dfrac{49}{4}\right)\sqrt{3}\right)\right)-\left(\left(-27\right)\sqrt{3}+\left(\dfrac{245}{6}\right)\sqrt{3}+\left(\dfrac{235}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{7}{2}\right)\sqrt{3}-11\right)-\left(\left(-\dfrac{2311}{36}\right)\sqrt{3}-\dfrac{77}{3}\right)\\
&=&\left(-\dfrac{7}{2}\right)\sqrt{3}-11+\left(\dfrac{2311}{36}\right)\sqrt{3}+\dfrac{77}{3}\\
&=&\left(\dfrac{2185}{36}\right)\sqrt{3}+\dfrac{44}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(1\right)\sqrt{75}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{4}{3}\right)\sqrt{9}\right)\right)-\left(\left(-2\right)\sqrt{12}-\dfrac{32}{3}+\left(\dfrac{68}{3}\right)\sqrt{27}+\dfrac{53}{3}\right)\right)\times\left(\left(\left(\left(-6\right)\sqrt{12}\right)-\left(\left(6\right)\sqrt{9}\right)-\dfrac{23}{3}-\left(\left(\dfrac{49}{8}\right)\sqrt{12}\right)\right)-\left(\left(-9\right)\sqrt{27}+\left(\dfrac{49}{6}\right)\sqrt{75}+\left(\dfrac{47}{9}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-\dfrac{111}{2}\right)\sqrt{3}\right)-4\right)-\left(\left(-4\right)\sqrt{3}-\dfrac{32}{3}+\left(68\right)\sqrt{3}+\dfrac{53}{3}\right)\right)\times\left(\left(\left(\left(-12\right)\sqrt{3}\right)-18-\dfrac{23}{3}-\left(\left(\dfrac{49}{4}\right)\sqrt{3}\right)\right)-\left(\left(-27\right)\sqrt{3}+\left(\dfrac{245}{6}\right)\sqrt{3}+\left(\dfrac{235}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{7}{2}\right)\sqrt{3}-11\right)\left(\left(-\dfrac{2311}{36}\right)\sqrt{3}-\dfrac{77}{3}\right)\\
&=&\left(\dfrac{16177}{72}\right)\sqrt{9}+\left(\dfrac{28655}{36}\right)\sqrt{3}+\dfrac{847}{3}\\
&=&\dfrac{7651}{8}+\left(\dfrac{28655}{36}\right)\sqrt{3}\\
\end{eqnarray*}