L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=9-\left(\left(-\dfrac{5}{4}\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{27}\right)\) et \( Y=\left(-\dfrac{13}{2}+\left(-\dfrac{23}{9}\right)\sqrt{27}-\dfrac{47}{4}\right)-\left(\left(\left(\dfrac{36}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{9}-\dfrac{11}{6}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(9-\left(\left(-\dfrac{5}{4}\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{27}\right)\right)+\left(\left(-\dfrac{13}{2}+\left(-\dfrac{23}{9}\right)\sqrt{27}-\dfrac{47}{4}\right)-\left(\left(\left(\dfrac{36}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{9}-\dfrac{11}{6}\right)\right)\\
&=&\left(9-\left(\left(-\dfrac{15}{4}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)+\left(\left(-\dfrac{13}{2}+\left(-\dfrac{23}{3}\right)\sqrt{3}-\dfrac{47}{4}\right)-\left(\left(\left(\dfrac{72}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{385}{4}\right)\sqrt{3}\right)\right)-\left(-\dfrac{39}{4}-\dfrac{11}{6}\right)\right)\\
&=&9-\left(\left(-\dfrac{15}{4}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)+\left(-\dfrac{13}{2}+\left(-\dfrac{23}{3}\right)\sqrt{3}-\dfrac{47}{4}\right)-\left(\left(\left(\dfrac{72}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{385}{4}\right)\sqrt{3}\right)\right)-\left(-\dfrac{39}{4}-\dfrac{11}{6}\right)\\
&=&\dfrac{7}{3}+\left(-\dfrac{3437}{30}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(9-\left(\left(-\dfrac{5}{4}\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{13}{2}+\left(-\dfrac{23}{9}\right)\sqrt{27}-\dfrac{47}{4}\right)-\left(\left(\left(\dfrac{36}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{9}-\dfrac{11}{6}\right)\right)\\
&=&\left(9-\left(\left(-\dfrac{15}{4}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{13}{2}+\left(-\dfrac{23}{3}\right)\sqrt{3}-\dfrac{47}{4}\right)-\left(\left(\left(\dfrac{72}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{385}{4}\right)\sqrt{3}\right)\right)-\left(-\dfrac{39}{4}-\dfrac{11}{6}\right)\right)\\
&=&\left(9+\left(\dfrac{15}{4}\right)\sqrt{3}\right)-\left(-\dfrac{20}{3}+\left(-\dfrac{7099}{60}\right)\sqrt{3}\right)\\
&=&9+\left(\dfrac{15}{4}\right)\sqrt{3}+\dfrac{20}{3}+\left(\dfrac{7099}{60}\right)\sqrt{3}\\
&=&\dfrac{47}{3}+\left(\dfrac{1831}{15}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(9-\left(\left(-\dfrac{5}{4}\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{27}\right)\right)\times\left(\left(-\dfrac{13}{2}+\left(-\dfrac{23}{9}\right)\sqrt{27}-\dfrac{47}{4}\right)-\left(\left(\left(\dfrac{36}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{9}-\dfrac{11}{6}\right)\right)\\
&=&\left(9-\left(\left(-\dfrac{15}{4}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)\times\left(\left(-\dfrac{13}{2}+\left(-\dfrac{23}{3}\right)\sqrt{3}-\dfrac{47}{4}\right)-\left(\left(\left(\dfrac{72}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{385}{4}\right)\sqrt{3}\right)\right)-\left(-\dfrac{39}{4}-\dfrac{11}{6}\right)\right)\\
&=&\left(9+\left(\dfrac{15}{4}\right)\sqrt{3}\right)\left(-\dfrac{20}{3}+\left(-\dfrac{7099}{60}\right)\sqrt{3}\right)\\
&=&-60+\left(-\dfrac{21797}{20}\right)\sqrt{3}+\left(-\dfrac{7099}{16}\right)\sqrt{9}\\
&=&-\dfrac{22257}{16}+\left(-\dfrac{21797}{20}\right)\sqrt{3}\\
\end{eqnarray*}