L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{64}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{27}{4}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{12}\right)-5+\left(-\dfrac{59}{8}\right)\sqrt{75}+\left(\left(-\dfrac{11}{2}\right)\sqrt{9}\right)-\dfrac{16}{3}-\left(\left(\dfrac{64}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{29}{9}\right)\sqrt{9}\right)\) et \( Y=\left(\left(0\right)\sqrt{75}\right)-\left(-\dfrac{15}{2}-\left(\left(\dfrac{1}{2}\right)\sqrt{12}\right)+\dfrac{49}{9}-\left(\left(\dfrac{67}{8}\right)\sqrt{27}\right)\right)-\dfrac{37}{3}-\left(\left(\left(-\dfrac{15}{8}\right)\sqrt{9}\right)-\dfrac{37}{3}-\left(\left(\dfrac{52}{9}\right)\sqrt{12}\right)\right)\) . 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(\dfrac{64}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{27}{4}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{12}\right)-5+\left(-\dfrac{59}{8}\right)\sqrt{75}+\left(\left(-\dfrac{11}{2}\right)\sqrt{9}\right)-\dfrac{16}{3}-\left(\left(\dfrac{64}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{29}{9}\right)\sqrt{9}\right)\right)+\left(\left(\left(0\right)\sqrt{75}\right)-\left(-\dfrac{15}{2}-\left(\left(\dfrac{1}{2}\right)\sqrt{12}\right)+\dfrac{49}{9}-\left(\left(\dfrac{67}{8}\right)\sqrt{27}\right)\right)-\dfrac{37}{3}-\left(\left(\left(-\dfrac{15}{8}\right)\sqrt{9}\right)-\dfrac{37}{3}-\left(\left(\dfrac{52}{9}\right)\sqrt{12}\right)\right)\right)\\
&=&\left(\left(\left(\dfrac{128}{9}\right)\sqrt{3}\right)+28+\dfrac{81}{4}-\left(\left(6\right)\sqrt{3}\right)-5+\left(-\dfrac{295}{8}\right)\sqrt{3}-\dfrac{33}{2}-\dfrac{16}{3}-\left(\left(64\right)\sqrt{3}\right)-\dfrac{29}{3}\right)+\left(\left(\left(0\right)\sqrt{3}\right)-\left(-\dfrac{15}{2}-\left(\left(1\right)\sqrt{3}\right)+\dfrac{49}{9}-\left(\left(\dfrac{201}{8}\right)\sqrt{3}\right)\right)-\dfrac{37}{3}-\left(-\dfrac{45}{8}-\dfrac{37}{3}-\left(\left(\dfrac{104}{9}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(\dfrac{128}{9}\right)\sqrt{3}\right)+28+\dfrac{81}{4}-\left(\left(6\right)\sqrt{3}\right)-5+\left(-\dfrac{295}{8}\right)\sqrt{3}-\dfrac{33}{2}-\dfrac{16}{3}-\left(\left(64\right)\sqrt{3}\right)-\dfrac{29}{3}+\left(\left(0\right)\sqrt{3}\right)-\left(-\dfrac{15}{2}-\left(\left(1\right)\sqrt{3}\right)+\dfrac{49}{9}-\left(\left(\dfrac{201}{8}\right)\sqrt{3}\right)\right)-\dfrac{37}{3}-\left(-\dfrac{45}{8}-\dfrac{37}{3}-\left(\left(\dfrac{104}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{1979}{36}\right)\sqrt{3}+\dfrac{1399}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{64}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{27}{4}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{12}\right)-5+\left(-\dfrac{59}{8}\right)\sqrt{75}+\left(\left(-\dfrac{11}{2}\right)\sqrt{9}\right)-\dfrac{16}{3}-\left(\left(\dfrac{64}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{29}{9}\right)\sqrt{9}\right)\right)-\left(\left(\left(0\right)\sqrt{75}\right)-\left(-\dfrac{15}{2}-\left(\left(\dfrac{1}{2}\right)\sqrt{12}\right)+\dfrac{49}{9}-\left(\left(\dfrac{67}{8}\right)\sqrt{27}\right)\right)-\dfrac{37}{3}-\left(\left(\left(-\dfrac{15}{8}\right)\sqrt{9}\right)-\dfrac{37}{3}-\left(\left(\dfrac{52}{9}\right)\sqrt{12}\right)\right)\right)\\
&=&\left(\left(\left(\dfrac{128}{9}\right)\sqrt{3}\right)+28+\dfrac{81}{4}-\left(\left(6\right)\sqrt{3}\right)-5+\left(-\dfrac{295}{8}\right)\sqrt{3}-\dfrac{33}{2}-\dfrac{16}{3}-\left(\left(64\right)\sqrt{3}\right)-\dfrac{29}{3}\right)-\left(\left(\left(0\right)\sqrt{3}\right)-\left(-\dfrac{15}{2}-\left(\left(1\right)\sqrt{3}\right)+\dfrac{49}{9}-\left(\left(\dfrac{201}{8}\right)\sqrt{3}\right)\right)-\dfrac{37}{3}-\left(-\dfrac{45}{8}-\dfrac{37}{3}-\left(\left(\dfrac{104}{9}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{6671}{72}\right)\sqrt{3}+\dfrac{47}{4}\right)-\left(\left(\dfrac{2713}{72}\right)\sqrt{3}+\dfrac{553}{72}\right)\\
&=&\left(-\dfrac{6671}{72}\right)\sqrt{3}+\dfrac{47}{4}+\left(-\dfrac{2713}{72}\right)\sqrt{3}-\dfrac{553}{72}\\
&=&\left(-\dfrac{391}{3}\right)\sqrt{3}+\dfrac{293}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{64}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{27}{4}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{12}\right)-5+\left(-\dfrac{59}{8}\right)\sqrt{75}+\left(\left(-\dfrac{11}{2}\right)\sqrt{9}\right)-\dfrac{16}{3}-\left(\left(\dfrac{64}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{29}{9}\right)\sqrt{9}\right)\right)\times\left(\left(\left(0\right)\sqrt{75}\right)-\left(-\dfrac{15}{2}-\left(\left(\dfrac{1}{2}\right)\sqrt{12}\right)+\dfrac{49}{9}-\left(\left(\dfrac{67}{8}\right)\sqrt{27}\right)\right)-\dfrac{37}{3}-\left(\left(\left(-\dfrac{15}{8}\right)\sqrt{9}\right)-\dfrac{37}{3}-\left(\left(\dfrac{52}{9}\right)\sqrt{12}\right)\right)\right)\\
&=&\left(\left(\left(\dfrac{128}{9}\right)\sqrt{3}\right)+28+\dfrac{81}{4}-\left(\left(6\right)\sqrt{3}\right)-5+\left(-\dfrac{295}{8}\right)\sqrt{3}-\dfrac{33}{2}-\dfrac{16}{3}-\left(\left(64\right)\sqrt{3}\right)-\dfrac{29}{3}\right)\times\left(\left(\left(0\right)\sqrt{3}\right)-\left(-\dfrac{15}{2}-\left(\left(1\right)\sqrt{3}\right)+\dfrac{49}{9}-\left(\left(\dfrac{201}{8}\right)\sqrt{3}\right)\right)-\dfrac{37}{3}-\left(-\dfrac{45}{8}-\dfrac{37}{3}-\left(\left(\dfrac{104}{9}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{6671}{72}\right)\sqrt{3}+\dfrac{47}{4}\right)\left(\left(\dfrac{2713}{72}\right)\sqrt{3}+\dfrac{553}{72}\right)\\
&=&\left(-\dfrac{18098423}{5184}\right)\sqrt{9}+\left(-\dfrac{1393865}{5184}\right)\sqrt{3}+\dfrac{25991}{288}\\
&=&-\dfrac{5167433376}{497664}+\left(-\dfrac{1393865}{5184}\right)\sqrt{3}\\
\end{eqnarray*}