L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{11}{4}+\left(-\dfrac{39}{2}\right)\sqrt{75}+\left(0\right)\sqrt{12}\right)-\left(-8+\left(\dfrac{1}{3}\right)\sqrt{75}+\left(\dfrac{53}{8}\right)\sqrt{9}\right)\) et \( Y=\left(\left(\left(\dfrac{13}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{66}{7}\right)\sqrt{27}\right)\right)-\left(\dfrac{32}{3}-\left(\left(-\dfrac{58}{7}\right)\sqrt{12}\right)\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{75}-\dfrac{71}{7}+\left(-\dfrac{1}{2}\right)\sqrt{75}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{11}{4}+\left(-\dfrac{39}{2}\right)\sqrt{75}+\left(0\right)\sqrt{12}\right)-\left(-8+\left(\dfrac{1}{3}\right)\sqrt{75}+\left(\dfrac{53}{8}\right)\sqrt{9}\right)\right)+\left(\left(\left(\left(\dfrac{13}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{66}{7}\right)\sqrt{27}\right)\right)-\left(\dfrac{32}{3}-\left(\left(-\dfrac{58}{7}\right)\sqrt{12}\right)\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{75}-\dfrac{71}{7}+\left(-\dfrac{1}{2}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\dfrac{11}{4}+\left(-\dfrac{195}{2}\right)\sqrt{3}+\left(0\right)\sqrt{3}\right)-\left(-8+\left(\dfrac{5}{3}\right)\sqrt{3}+\dfrac{159}{8}\right)\right)+\left(\left(\left(\left(\dfrac{65}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{165}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{198}{7}\right)\sqrt{3}\right)\right)-\left(\dfrac{32}{3}-\left(\left(-\dfrac{116}{7}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{165}{7}\right)\sqrt{3}-\dfrac{71}{7}+\left(-\dfrac{5}{2}\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{11}{4}+\left(-\dfrac{195}{2}\right)\sqrt{3}+\left(0\right)\sqrt{3}\right)-\left(-8+\left(\dfrac{5}{3}\right)\sqrt{3}+\dfrac{159}{8}\right)+\left(\left(\left(\dfrac{65}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{165}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{198}{7}\right)\sqrt{3}\right)\right)-\left(\dfrac{32}{3}-\left(\left(-\dfrac{116}{7}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{165}{7}\right)\sqrt{3}-\dfrac{71}{7}+\left(-\dfrac{5}{2}\right)\sqrt{3}\right)\\
&=&-\dfrac{1621}{168}+\left(-\dfrac{773}{7}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{11}{4}+\left(-\dfrac{39}{2}\right)\sqrt{75}+\left(0\right)\sqrt{12}\right)-\left(-8+\left(\dfrac{1}{3}\right)\sqrt{75}+\left(\dfrac{53}{8}\right)\sqrt{9}\right)\right)-\left(\left(\left(\left(\dfrac{13}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{66}{7}\right)\sqrt{27}\right)\right)-\left(\dfrac{32}{3}-\left(\left(-\dfrac{58}{7}\right)\sqrt{12}\right)\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{75}-\dfrac{71}{7}+\left(-\dfrac{1}{2}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\dfrac{11}{4}+\left(-\dfrac{195}{2}\right)\sqrt{3}+\left(0\right)\sqrt{3}\right)-\left(-8+\left(\dfrac{5}{3}\right)\sqrt{3}+\dfrac{159}{8}\right)\right)-\left(\left(\left(\left(\dfrac{65}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{165}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{198}{7}\right)\sqrt{3}\right)\right)-\left(\dfrac{32}{3}-\left(\left(-\dfrac{116}{7}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{165}{7}\right)\sqrt{3}-\dfrac{71}{7}+\left(-\dfrac{5}{2}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{73}{8}+\left(-\dfrac{595}{6}\right)\sqrt{3}\right)-\left(\left(-\dfrac{473}{42}\right)\sqrt{3}-\dfrac{11}{21}\right)\\
&=&-\dfrac{73}{8}+\left(-\dfrac{595}{6}\right)\sqrt{3}+\left(\dfrac{473}{42}\right)\sqrt{3}+\dfrac{11}{21}\\
&=&-\dfrac{1445}{168}+\left(-\dfrac{1846}{21}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{11}{4}+\left(-\dfrac{39}{2}\right)\sqrt{75}+\left(0\right)\sqrt{12}\right)-\left(-8+\left(\dfrac{1}{3}\right)\sqrt{75}+\left(\dfrac{53}{8}\right)\sqrt{9}\right)\right)\times\left(\left(\left(\left(\dfrac{13}{3}\right)\sqrt{75}\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{66}{7}\right)\sqrt{27}\right)\right)-\left(\dfrac{32}{3}-\left(\left(-\dfrac{58}{7}\right)\sqrt{12}\right)\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{75}-\dfrac{71}{7}+\left(-\dfrac{1}{2}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\dfrac{11}{4}+\left(-\dfrac{195}{2}\right)\sqrt{3}+\left(0\right)\sqrt{3}\right)-\left(-8+\left(\dfrac{5}{3}\right)\sqrt{3}+\dfrac{159}{8}\right)\right)\times\left(\left(\left(\left(\dfrac{65}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{165}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{198}{7}\right)\sqrt{3}\right)\right)-\left(\dfrac{32}{3}-\left(\left(-\dfrac{116}{7}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{165}{7}\right)\sqrt{3}-\dfrac{71}{7}+\left(-\dfrac{5}{2}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{73}{8}+\left(-\dfrac{595}{6}\right)\sqrt{3}\right)\left(\left(-\dfrac{473}{42}\right)\sqrt{3}-\dfrac{11}{21}\right)\\
&=&\left(\dfrac{155947}{1008}\right)\sqrt{3}+\dfrac{803}{168}+\left(\dfrac{40205}{36}\right)\sqrt{9}\\
&=&\left(\dfrac{155947}{1008}\right)\sqrt{3}+\dfrac{187891}{56}\\
\end{eqnarray*}