L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{40}{3}\right)\sqrt{50}+\left(-\dfrac{26}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{8}\right)-\left(4+\left(-\dfrac{64}{3}\right)\sqrt{4}+\left(4\right)\sqrt{18}\right)\) et \( Y=\dfrac{28}{5}-\dfrac{12}{5}-2-\dfrac{43}{8}-\left(\left(\dfrac{32}{3}\right)\sqrt{18}\right)-2+\left(\left(\dfrac{33}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{79}{2}\right)\sqrt{4}\right)-\left(\left(0\right)\sqrt{8}\right)\) . 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{40}{3}\right)\sqrt{50}+\left(-\dfrac{26}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{8}\right)-\left(4+\left(-\dfrac{64}{3}\right)\sqrt{4}+\left(4\right)\sqrt{18}\right)\right)+\left(\dfrac{28}{5}-\dfrac{12}{5}-2-\dfrac{43}{8}-\left(\left(\dfrac{32}{3}\right)\sqrt{18}\right)-2+\left(\left(\dfrac{33}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{79}{2}\right)\sqrt{4}\right)-\left(\left(0\right)\sqrt{8}\right)\right)\\
&=&\left(\left(\left(-\dfrac{200}{3}\right)\sqrt{2}-\dfrac{52}{3}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{2}\right)-\left(4-\dfrac{128}{3}+\left(12\right)\sqrt{2}\right)\right)+\left(\dfrac{28}{5}-\dfrac{12}{5}-2-\dfrac{43}{8}-\left(\left(32\right)\sqrt{2}\right)-2+\left(\left(\dfrac{99}{4}\right)\sqrt{2}\right)+79-\left(\left(0\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{200}{3}\right)\sqrt{2}-\dfrac{52}{3}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{2}\right)-\left(4-\dfrac{128}{3}+\left(12\right)\sqrt{2}\right)+\dfrac{28}{5}-\dfrac{12}{5}-2-\dfrac{43}{8}-\left(\left(32\right)\sqrt{2}\right)-2+\left(\left(\dfrac{99}{4}\right)\sqrt{2}\right)+79-\left(\left(0\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{3301}{36}\right)\sqrt{2}+\dfrac{11299}{120}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{40}{3}\right)\sqrt{50}+\left(-\dfrac{26}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{8}\right)-\left(4+\left(-\dfrac{64}{3}\right)\sqrt{4}+\left(4\right)\sqrt{18}\right)\right)-\left(\dfrac{28}{5}-\dfrac{12}{5}-2-\dfrac{43}{8}-\left(\left(\dfrac{32}{3}\right)\sqrt{18}\right)-2+\left(\left(\dfrac{33}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{79}{2}\right)\sqrt{4}\right)-\left(\left(0\right)\sqrt{8}\right)\right)\\
&=&\left(\left(\left(-\dfrac{200}{3}\right)\sqrt{2}-\dfrac{52}{3}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{2}\right)-\left(4-\dfrac{128}{3}+\left(12\right)\sqrt{2}\right)\right)-\left(\dfrac{28}{5}-\dfrac{12}{5}-2-\dfrac{43}{8}-\left(\left(32\right)\sqrt{2}\right)-2+\left(\left(\dfrac{99}{4}\right)\sqrt{2}\right)+79-\left(\left(0\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{760}{9}\right)\sqrt{2}+\dfrac{64}{3}\right)-\left(\dfrac{2913}{40}+\left(-\dfrac{29}{4}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{760}{9}\right)\sqrt{2}+\dfrac{64}{3}+-\dfrac{2913}{40}+\left(\dfrac{29}{4}\right)\sqrt{2}\\
&=&\left(-\dfrac{2779}{36}\right)\sqrt{2}-\dfrac{6179}{120}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{40}{3}\right)\sqrt{50}+\left(-\dfrac{26}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{8}\right)-\left(4+\left(-\dfrac{64}{3}\right)\sqrt{4}+\left(4\right)\sqrt{18}\right)\right)\times\left(\dfrac{28}{5}-\dfrac{12}{5}-2-\dfrac{43}{8}-\left(\left(\dfrac{32}{3}\right)\sqrt{18}\right)-2+\left(\left(\dfrac{33}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{79}{2}\right)\sqrt{4}\right)-\left(\left(0\right)\sqrt{8}\right)\right)\\
&=&\left(\left(\left(-\dfrac{200}{3}\right)\sqrt{2}-\dfrac{52}{3}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{2}\right)-\left(4-\dfrac{128}{3}+\left(12\right)\sqrt{2}\right)\right)\times\left(\dfrac{28}{5}-\dfrac{12}{5}-2-\dfrac{43}{8}-\left(\left(32\right)\sqrt{2}\right)-2+\left(\left(\dfrac{99}{4}\right)\sqrt{2}\right)+79-\left(\left(0\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{760}{9}\right)\sqrt{2}+\dfrac{64}{3}\right)\left(\dfrac{2913}{40}+\left(-\dfrac{29}{4}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{18913}{3}\right)\sqrt{2}+\left(\dfrac{5510}{9}\right)\sqrt{4}+\dfrac{7768}{5}\\
&=&\left(-\dfrac{18913}{3}\right)\sqrt{2}+\dfrac{125012}{45}\\
\end{eqnarray*}