L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(-9\right)\sqrt{12}+\dfrac{13}{4}-\dfrac{44}{9}-\left(\left(-\dfrac{15}{2}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{9}\right)+\left(\dfrac{39}{4}\right)\sqrt{75}+\dfrac{36}{5}-\left(\left(8\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{16}{3}\right)\sqrt{75}\right)-\left(\left(7\right)\sqrt{75}\right)\) et \( Y=\left(-\dfrac{9}{4}+\left(\dfrac{44}{7}\right)\sqrt{12}+6\right)-\left(\left(\left(\dfrac{8}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{12}{7}\right)\sqrt{75}\right)+\dfrac{1}{3}\right)-\left(\left(\left(-9\right)\sqrt{75}\right)-\dfrac{78}{5}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-9\right)\sqrt{12}+\dfrac{13}{4}-\dfrac{44}{9}-\left(\left(-\dfrac{15}{2}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{9}\right)+\left(\dfrac{39}{4}\right)\sqrt{75}+\dfrac{36}{5}-\left(\left(8\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{16}{3}\right)\sqrt{75}\right)-\left(\left(7\right)\sqrt{75}\right)\right)+\left(\left(-\dfrac{9}{4}+\left(\dfrac{44}{7}\right)\sqrt{12}+6\right)-\left(\left(\left(\dfrac{8}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{12}{7}\right)\sqrt{75}\right)+\dfrac{1}{3}\right)-\left(\left(\left(-9\right)\sqrt{75}\right)-\dfrac{78}{5}\right)\right)\\
&=&\left(\left(-18\right)\sqrt{3}+\dfrac{13}{4}-\dfrac{44}{9}-\left(\left(-15\right)\sqrt{3}\right)+12+\left(\dfrac{195}{4}\right)\sqrt{3}+\dfrac{36}{5}-\left(\left(16\right)\sqrt{3}\right)-\left(\left(19\right)\sqrt{3}\right)-\left(\left(\dfrac{80}{3}\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)\right)+\left(\left(-\dfrac{9}{4}+\left(\dfrac{88}{7}\right)\sqrt{3}+6\right)-\left(\left(\left(8\right)\sqrt{3}\right)-\left(\left(\dfrac{60}{7}\right)\sqrt{3}\right)+\dfrac{1}{3}\right)-\left(\left(\left(-45\right)\sqrt{3}\right)-\dfrac{78}{5}\right)\right)\\
&=&\left(-18\right)\sqrt{3}+\dfrac{13}{4}-\dfrac{44}{9}-\left(\left(-15\right)\sqrt{3}\right)+12+\left(\dfrac{195}{4}\right)\sqrt{3}+\dfrac{36}{5}-\left(\left(16\right)\sqrt{3}\right)-\left(\left(19\right)\sqrt{3}\right)-\left(\left(\dfrac{80}{3}\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)+\left(-\dfrac{9}{4}+\left(\dfrac{88}{7}\right)\sqrt{3}+6\right)-\left(\left(\left(8\right)\sqrt{3}\right)-\left(\left(\dfrac{60}{7}\right)\sqrt{3}\right)+\dfrac{1}{3}\right)-\left(\left(\left(-45\right)\sqrt{3}\right)-\dfrac{78}{5}\right)\\
&=&\left(\dfrac{607}{84}\right)\sqrt{3}+\dfrac{1646}{45}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-9\right)\sqrt{12}+\dfrac{13}{4}-\dfrac{44}{9}-\left(\left(-\dfrac{15}{2}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{9}\right)+\left(\dfrac{39}{4}\right)\sqrt{75}+\dfrac{36}{5}-\left(\left(8\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{16}{3}\right)\sqrt{75}\right)-\left(\left(7\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{9}{4}+\left(\dfrac{44}{7}\right)\sqrt{12}+6\right)-\left(\left(\left(\dfrac{8}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{12}{7}\right)\sqrt{75}\right)+\dfrac{1}{3}\right)-\left(\left(\left(-9\right)\sqrt{75}\right)-\dfrac{78}{5}\right)\right)\\
&=&\left(\left(-18\right)\sqrt{3}+\dfrac{13}{4}-\dfrac{44}{9}-\left(\left(-15\right)\sqrt{3}\right)+12+\left(\dfrac{195}{4}\right)\sqrt{3}+\dfrac{36}{5}-\left(\left(16\right)\sqrt{3}\right)-\left(\left(19\right)\sqrt{3}\right)-\left(\left(\dfrac{80}{3}\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{9}{4}+\left(\dfrac{88}{7}\right)\sqrt{3}+6\right)-\left(\left(\left(8\right)\sqrt{3}\right)-\left(\left(\dfrac{60}{7}\right)\sqrt{3}\right)+\dfrac{1}{3}\right)-\left(\left(\left(-45\right)\sqrt{3}\right)-\dfrac{78}{5}\right)\right)\\
&=&\left(\left(-\dfrac{611}{12}\right)\sqrt{3}+\dfrac{3161}{180}\right)-\left(\dfrac{1141}{60}+\left(\dfrac{407}{7}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{611}{12}\right)\sqrt{3}+\dfrac{3161}{180}+-\dfrac{1141}{60}+\left(-\dfrac{407}{7}\right)\sqrt{3}\\
&=&\left(-\dfrac{9161}{84}\right)\sqrt{3}-\dfrac{131}{90}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-9\right)\sqrt{12}+\dfrac{13}{4}-\dfrac{44}{9}-\left(\left(-\dfrac{15}{2}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{9}\right)+\left(\dfrac{39}{4}\right)\sqrt{75}+\dfrac{36}{5}-\left(\left(8\right)\sqrt{12}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{27}\right)-\left(\left(\dfrac{16}{3}\right)\sqrt{75}\right)-\left(\left(7\right)\sqrt{75}\right)\right)\times\left(\left(-\dfrac{9}{4}+\left(\dfrac{44}{7}\right)\sqrt{12}+6\right)-\left(\left(\left(\dfrac{8}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{12}{7}\right)\sqrt{75}\right)+\dfrac{1}{3}\right)-\left(\left(\left(-9\right)\sqrt{75}\right)-\dfrac{78}{5}\right)\right)\\
&=&\left(\left(-18\right)\sqrt{3}+\dfrac{13}{4}-\dfrac{44}{9}-\left(\left(-15\right)\sqrt{3}\right)+12+\left(\dfrac{195}{4}\right)\sqrt{3}+\dfrac{36}{5}-\left(\left(16\right)\sqrt{3}\right)-\left(\left(19\right)\sqrt{3}\right)-\left(\left(\dfrac{80}{3}\right)\sqrt{3}\right)-\left(\left(35\right)\sqrt{3}\right)\right)\times\left(\left(-\dfrac{9}{4}+\left(\dfrac{88}{7}\right)\sqrt{3}+6\right)-\left(\left(\left(8\right)\sqrt{3}\right)-\left(\left(\dfrac{60}{7}\right)\sqrt{3}\right)+\dfrac{1}{3}\right)-\left(\left(\left(-45\right)\sqrt{3}\right)-\dfrac{78}{5}\right)\right)\\
&=&\left(\left(-\dfrac{611}{12}\right)\sqrt{3}+\dfrac{3161}{180}\right)\left(\dfrac{1141}{60}+\left(\dfrac{407}{7}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{266051}{5040}\right)\sqrt{3}+\left(-\dfrac{248677}{84}\right)\sqrt{9}+\dfrac{3606701}{10800}\\
&=&\left(\dfrac{266051}{5040}\right)\sqrt{3}-\dfrac{2584723972}{302400}\\
\end{eqnarray*}