L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\left(8\right)\sqrt{27}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{27}\right)\right)-\left(\left(\left(-\dfrac{22}{9}\right)\sqrt{75}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{9}\right)-\left(\left(5\right)\sqrt{75}\right)\right)-\left(\left(\left(-1\right)\sqrt{9}\right)+8-\left(\left(\dfrac{35}{2}\right)\sqrt{12}\right)\right)\) et \( Y=\left(\left(-3\right)\sqrt{9}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{75}+\left(6\right)\sqrt{27}-\dfrac{26}{5}-\dfrac{13}{9}\right)-\left(\left(-\dfrac{19}{5}\right)\sqrt{75}+\left(\dfrac{64}{5}\right)\sqrt{75}+\left(-4\right)\sqrt{75}+\left(-1\right)\sqrt{9}\right)\) . 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(8\right)\sqrt{27}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{27}\right)\right)-\left(\left(\left(-\dfrac{22}{9}\right)\sqrt{75}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{9}\right)-\left(\left(5\right)\sqrt{75}\right)\right)-\left(\left(\left(-1\right)\sqrt{9}\right)+8-\left(\left(\dfrac{35}{2}\right)\sqrt{12}\right)\right)\right)+\left(\left(\left(-3\right)\sqrt{9}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{75}+\left(6\right)\sqrt{27}-\dfrac{26}{5}-\dfrac{13}{9}\right)-\left(\left(-\dfrac{19}{5}\right)\sqrt{75}+\left(\dfrac{64}{5}\right)\sqrt{75}+\left(-4\right)\sqrt{75}+\left(-1\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(\left(24\right)\sqrt{3}\right)-\left(\left(\dfrac{162}{7}\right)\sqrt{3}\right)\right)-\left(\left(\left(-\dfrac{110}{9}\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)+\dfrac{231}{8}-\left(\left(25\right)\sqrt{3}\right)\right)-\left(-3+8-\left(\left(35\right)\sqrt{3}\right)\right)\right)+\left(-9-\dfrac{45}{4}-\left(\left(\dfrac{130}{3}\right)\sqrt{3}+\left(18\right)\sqrt{3}-\dfrac{26}{5}-\dfrac{13}{9}\right)-\left(\left(-19\right)\sqrt{3}+\left(64\right)\sqrt{3}+\left(-20\right)\sqrt{3}-3\right)\right)\\
&=&\left(\left(\left(24\right)\sqrt{3}\right)-\left(\left(\dfrac{162}{7}\right)\sqrt{3}\right)\right)-\left(\left(\left(-\dfrac{110}{9}\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)+\dfrac{231}{8}-\left(\left(25\right)\sqrt{3}\right)\right)-\left(-3+8-\left(\left(35\right)\sqrt{3}\right)\right)-9-\dfrac{45}{4}-\left(\left(\dfrac{130}{3}\right)\sqrt{3}+\left(18\right)\sqrt{3}-\dfrac{26}{5}-\dfrac{13}{9}\right)-\left(\left(-19\right)\sqrt{3}+\left(64\right)\sqrt{3}+\left(-20\right)\sqrt{3}-3\right)\\
&=&\left(\dfrac{1921}{126}\right)\sqrt{3}-\dfrac{16013}{360}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(8\right)\sqrt{27}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{27}\right)\right)-\left(\left(\left(-\dfrac{22}{9}\right)\sqrt{75}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{9}\right)-\left(\left(5\right)\sqrt{75}\right)\right)-\left(\left(\left(-1\right)\sqrt{9}\right)+8-\left(\left(\dfrac{35}{2}\right)\sqrt{12}\right)\right)\right)-\left(\left(\left(-3\right)\sqrt{9}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{75}+\left(6\right)\sqrt{27}-\dfrac{26}{5}-\dfrac{13}{9}\right)-\left(\left(-\dfrac{19}{5}\right)\sqrt{75}+\left(\dfrac{64}{5}\right)\sqrt{75}+\left(-4\right)\sqrt{75}+\left(-1\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(\left(24\right)\sqrt{3}\right)-\left(\left(\dfrac{162}{7}\right)\sqrt{3}\right)\right)-\left(\left(\left(-\dfrac{110}{9}\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)+\dfrac{231}{8}-\left(\left(25\right)\sqrt{3}\right)\right)-\left(-3+8-\left(\left(35\right)\sqrt{3}\right)\right)\right)-\left(-9-\dfrac{45}{4}-\left(\left(\dfrac{130}{3}\right)\sqrt{3}+\left(18\right)\sqrt{3}-\dfrac{26}{5}-\dfrac{13}{9}\right)-\left(\left(-19\right)\sqrt{3}+\left(64\right)\sqrt{3}+\left(-20\right)\sqrt{3}-3\right)\right)\\
&=&\left(\left(\dfrac{12799}{126}\right)\sqrt{3}-\dfrac{271}{8}\right)-\left(-\dfrac{1909}{180}+\left(-\dfrac{259}{3}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{12799}{126}\right)\sqrt{3}-\dfrac{271}{8}+\dfrac{1909}{180}+\left(\dfrac{259}{3}\right)\sqrt{3}\\
&=&\left(\dfrac{23677}{126}\right)\sqrt{3}-\dfrac{8377}{360}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(8\right)\sqrt{27}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{27}\right)\right)-\left(\left(\left(-\dfrac{22}{9}\right)\sqrt{75}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{9}\right)-\left(\left(5\right)\sqrt{75}\right)\right)-\left(\left(\left(-1\right)\sqrt{9}\right)+8-\left(\left(\dfrac{35}{2}\right)\sqrt{12}\right)\right)\right)\times\left(\left(\left(-3\right)\sqrt{9}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{75}+\left(6\right)\sqrt{27}-\dfrac{26}{5}-\dfrac{13}{9}\right)-\left(\left(-\dfrac{19}{5}\right)\sqrt{75}+\left(\dfrac{64}{5}\right)\sqrt{75}+\left(-4\right)\sqrt{75}+\left(-1\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(\left(24\right)\sqrt{3}\right)-\left(\left(\dfrac{162}{7}\right)\sqrt{3}\right)\right)-\left(\left(\left(-\dfrac{110}{9}\right)\sqrt{3}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{3}\right)+\dfrac{231}{8}-\left(\left(25\right)\sqrt{3}\right)\right)-\left(-3+8-\left(\left(35\right)\sqrt{3}\right)\right)\right)\times\left(-9-\dfrac{45}{4}-\left(\left(\dfrac{130}{3}\right)\sqrt{3}+\left(18\right)\sqrt{3}-\dfrac{26}{5}-\dfrac{13}{9}\right)-\left(\left(-19\right)\sqrt{3}+\left(64\right)\sqrt{3}+\left(-20\right)\sqrt{3}-3\right)\right)\\
&=&\left(\left(\dfrac{12799}{126}\right)\sqrt{3}-\dfrac{271}{8}\right)\left(-\dfrac{1909}{180}+\left(-\dfrac{259}{3}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{20947657}{11340}\right)\sqrt{3}+\left(-\dfrac{473563}{54}\right)\sqrt{9}+\dfrac{517339}{1440}\\
&=&\left(\dfrac{20947657}{11340}\right)\sqrt{3}-\dfrac{37367701}{1440}\\
\end{eqnarray*}