L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\left(-7\right)\sqrt{75}\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{27}\right)-\left(-\dfrac{79}{7}+\left(\dfrac{26}{3}\right)\sqrt{75}+\left(-\dfrac{8}{9}\right)\sqrt{27}\right)\) et \( Y=\left(\dfrac{21}{4}\right)\sqrt{9}-3+\dfrac{71}{2}+\left(-\dfrac{64}{3}\right)\sqrt{27}+\left(9\right)\sqrt{12}+\left(-\dfrac{58}{5}\right)\sqrt{12}+\dfrac{61}{7}+\left(-\dfrac{5}{3}\right)\sqrt{12}+\left(\dfrac{32}{3}\right)\sqrt{9}-4+\dfrac{23}{4}+\left(-4\right)\sqrt{9}+\left(-\dfrac{25}{7}\right)\sqrt{75}\) . 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(-7\right)\sqrt{75}\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{27}\right)-\left(-\dfrac{79}{7}+\left(\dfrac{26}{3}\right)\sqrt{75}+\left(-\dfrac{8}{9}\right)\sqrt{27}\right)\right)+\left(\left(\dfrac{21}{4}\right)\sqrt{9}-3+\dfrac{71}{2}+\left(-\dfrac{64}{3}\right)\sqrt{27}+\left(9\right)\sqrt{12}+\left(-\dfrac{58}{5}\right)\sqrt{12}+\dfrac{61}{7}+\left(-\dfrac{5}{3}\right)\sqrt{12}+\left(\dfrac{32}{3}\right)\sqrt{9}-4+\dfrac{23}{4}+\left(-4\right)\sqrt{9}+\left(-\dfrac{25}{7}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(\left(-35\right)\sqrt{3}\right)-\left(\left(\dfrac{219}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{192}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{360}{7}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{219}{5}\right)\sqrt{3}\right)-\left(-\dfrac{79}{7}+\left(\dfrac{130}{3}\right)\sqrt{3}+\left(-\dfrac{8}{3}\right)\sqrt{3}\right)\right)+\left(\dfrac{63}{4}-3+\dfrac{71}{2}+\left(-64\right)\sqrt{3}+\left(18\right)\sqrt{3}+\left(-\dfrac{116}{5}\right)\sqrt{3}+\dfrac{61}{7}+\left(-\dfrac{10}{3}\right)\sqrt{3}+32-4+\dfrac{23}{4}-12+\left(-\dfrac{125}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(\left(-35\right)\sqrt{3}\right)-\left(\left(\dfrac{219}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{192}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{360}{7}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{219}{5}\right)\sqrt{3}\right)-\left(-\dfrac{79}{7}+\left(\dfrac{130}{3}\right)\sqrt{3}+\left(-\dfrac{8}{3}\right)\sqrt{3}\right)+\dfrac{63}{4}-3+\dfrac{71}{2}+\left(-64\right)\sqrt{3}+\left(18\right)\sqrt{3}+\left(-\dfrac{116}{5}\right)\sqrt{3}+\dfrac{61}{7}+\left(-\dfrac{10}{3}\right)\sqrt{3}+32-4+\dfrac{23}{4}-12+\left(-\dfrac{125}{7}\right)\sqrt{3}\\
&=&\left(-\dfrac{11638}{35}\right)\sqrt{3}+90\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-7\right)\sqrt{75}\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{27}\right)-\left(-\dfrac{79}{7}+\left(\dfrac{26}{3}\right)\sqrt{75}+\left(-\dfrac{8}{9}\right)\sqrt{27}\right)\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{9}-3+\dfrac{71}{2}+\left(-\dfrac{64}{3}\right)\sqrt{27}+\left(9\right)\sqrt{12}+\left(-\dfrac{58}{5}\right)\sqrt{12}+\dfrac{61}{7}+\left(-\dfrac{5}{3}\right)\sqrt{12}+\left(\dfrac{32}{3}\right)\sqrt{9}-4+\dfrac{23}{4}+\left(-4\right)\sqrt{9}+\left(-\dfrac{25}{7}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(\left(-35\right)\sqrt{3}\right)-\left(\left(\dfrac{219}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{192}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{360}{7}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{219}{5}\right)\sqrt{3}\right)-\left(-\dfrac{79}{7}+\left(\dfrac{130}{3}\right)\sqrt{3}+\left(-\dfrac{8}{3}\right)\sqrt{3}\right)\right)-\left(\dfrac{63}{4}-3+\dfrac{71}{2}+\left(-64\right)\sqrt{3}+\left(18\right)\sqrt{3}+\left(-\dfrac{116}{5}\right)\sqrt{3}+\dfrac{61}{7}+\left(-\dfrac{10}{3}\right)\sqrt{3}+32-4+\dfrac{23}{4}-12+\left(-\dfrac{125}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{25423}{105}\right)\sqrt{3}+\dfrac{79}{7}\right)-\left(\dfrac{551}{7}+\left(-\dfrac{9491}{105}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{25423}{105}\right)\sqrt{3}+\dfrac{79}{7}+-\dfrac{551}{7}+\left(\dfrac{9491}{105}\right)\sqrt{3}\\
&=&\left(-\dfrac{2276}{15}\right)\sqrt{3}-\dfrac{472}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-7\right)\sqrt{75}\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{27}\right)-\left(-\dfrac{79}{7}+\left(\dfrac{26}{3}\right)\sqrt{75}+\left(-\dfrac{8}{9}\right)\sqrt{27}\right)\right)\times\left(\left(\dfrac{21}{4}\right)\sqrt{9}-3+\dfrac{71}{2}+\left(-\dfrac{64}{3}\right)\sqrt{27}+\left(9\right)\sqrt{12}+\left(-\dfrac{58}{5}\right)\sqrt{12}+\dfrac{61}{7}+\left(-\dfrac{5}{3}\right)\sqrt{12}+\left(\dfrac{32}{3}\right)\sqrt{9}-4+\dfrac{23}{4}+\left(-4\right)\sqrt{9}+\left(-\dfrac{25}{7}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(\left(-35\right)\sqrt{3}\right)-\left(\left(\dfrac{219}{5}\right)\sqrt{3}\right)-\left(\left(\dfrac{192}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{360}{7}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{219}{5}\right)\sqrt{3}\right)-\left(-\dfrac{79}{7}+\left(\dfrac{130}{3}\right)\sqrt{3}+\left(-\dfrac{8}{3}\right)\sqrt{3}\right)\right)\times\left(\dfrac{63}{4}-3+\dfrac{71}{2}+\left(-64\right)\sqrt{3}+\left(18\right)\sqrt{3}+\left(-\dfrac{116}{5}\right)\sqrt{3}+\dfrac{61}{7}+\left(-\dfrac{10}{3}\right)\sqrt{3}+32-4+\dfrac{23}{4}-12+\left(-\dfrac{125}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{25423}{105}\right)\sqrt{3}+\dfrac{79}{7}\right)\left(\dfrac{551}{7}+\left(-\dfrac{9491}{105}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{10847028570}{540225}\right)\sqrt{3}+\left(\dfrac{241289693}{11025}\right)\sqrt{9}+\dfrac{43529}{49}\\
&=&\left(-\dfrac{10847028570}{540225}\right)\sqrt{3}+\dfrac{11983164032}{180075}\\
\end{eqnarray*}