L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{31}{5}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(\dfrac{52}{3}\right)\sqrt{12}+\dfrac{52}{3}+9\right)-\left(9-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{12}\right)\right)\) et \( Y=\left(\left(\dfrac{11}{4}\right)\sqrt{27}\right)+\dfrac{1}{3}-\left(\left(4\right)\sqrt{27}\right)-3+\left(\dfrac{78}{5}\right)\sqrt{27}\) . 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{31}{5}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(\dfrac{52}{3}\right)\sqrt{12}+\dfrac{52}{3}+9\right)-\left(9-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{12}\right)\right)\right)+\left(\left(\left(\dfrac{11}{4}\right)\sqrt{27}\right)+\dfrac{1}{3}-\left(\left(4\right)\sqrt{27}\right)-3+\left(\dfrac{78}{5}\right)\sqrt{27}\right)\\
&=&\left(\left(\left(-\dfrac{93}{5}\right)\sqrt{3}\right)-3-\left(\left(\dfrac{104}{3}\right)\sqrt{3}+\dfrac{52}{3}+9\right)-\left(9-0-\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{3}\right)\right)\right)+\left(\left(\left(\dfrac{33}{4}\right)\sqrt{3}\right)+\dfrac{1}{3}-\left(\left(12\right)\sqrt{3}\right)-3+\left(\dfrac{234}{5}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{93}{5}\right)\sqrt{3}\right)-3-\left(\left(\dfrac{104}{3}\right)\sqrt{3}+\dfrac{52}{3}+9\right)-\left(9-0-\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{3}\right)\right)+\left(\left(\dfrac{33}{4}\right)\sqrt{3}\right)+\dfrac{1}{3}-\left(\left(12\right)\sqrt{3}\right)-3+\left(\dfrac{234}{5}\right)\sqrt{3}\\
&=&\left(-\dfrac{857}{12}\right)\sqrt{3}-41\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{31}{5}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(\dfrac{52}{3}\right)\sqrt{12}+\dfrac{52}{3}+9\right)-\left(9-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{12}\right)\right)\right)-\left(\left(\left(\dfrac{11}{4}\right)\sqrt{27}\right)+\dfrac{1}{3}-\left(\left(4\right)\sqrt{27}\right)-3+\left(\dfrac{78}{5}\right)\sqrt{27}\right)\\
&=&\left(\left(\left(-\dfrac{93}{5}\right)\sqrt{3}\right)-3-\left(\left(\dfrac{104}{3}\right)\sqrt{3}+\dfrac{52}{3}+9\right)-\left(9-0-\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{3}\right)\right)\right)-\left(\left(\left(\dfrac{33}{4}\right)\sqrt{3}\right)+\dfrac{1}{3}-\left(\left(12\right)\sqrt{3}\right)-3+\left(\dfrac{234}{5}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{1717}{15}\right)\sqrt{3}-\dfrac{115}{3}\right)-\left(\left(\dfrac{861}{20}\right)\sqrt{3}-\dfrac{8}{3}\right)\\
&=&\left(-\dfrac{1717}{15}\right)\sqrt{3}-\dfrac{115}{3}+\left(-\dfrac{861}{20}\right)\sqrt{3}+\dfrac{8}{3}\\
&=&\left(-\dfrac{9451}{60}\right)\sqrt{3}-\dfrac{107}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{31}{5}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(\dfrac{52}{3}\right)\sqrt{12}+\dfrac{52}{3}+9\right)-\left(9-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{12}\right)\right)\right)\times\left(\left(\left(\dfrac{11}{4}\right)\sqrt{27}\right)+\dfrac{1}{3}-\left(\left(4\right)\sqrt{27}\right)-3+\left(\dfrac{78}{5}\right)\sqrt{27}\right)\\
&=&\left(\left(\left(-\dfrac{93}{5}\right)\sqrt{3}\right)-3-\left(\left(\dfrac{104}{3}\right)\sqrt{3}+\dfrac{52}{3}+9\right)-\left(9-0-\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{3}\right)\right)\right)\times\left(\left(\left(\dfrac{33}{4}\right)\sqrt{3}\right)+\dfrac{1}{3}-\left(\left(12\right)\sqrt{3}\right)-3+\left(\dfrac{234}{5}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{1717}{15}\right)\sqrt{3}-\dfrac{115}{3}\right)\left(\left(\dfrac{861}{20}\right)\sqrt{3}-\dfrac{8}{3}\right)\\
&=&\left(-\dfrac{492779}{100}\right)\sqrt{9}+\left(-\dfrac{242101}{180}\right)\sqrt{3}+\dfrac{920}{9}\\
&=&-\dfrac{13213033}{900}+\left(-\dfrac{242101}{180}\right)\sqrt{3}\\
\end{eqnarray*}