L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\dfrac{52}{5}\right)\sqrt{45}+\left(3\right)\sqrt{45}+\left(-9\right)\sqrt{20}\right)-\left(-4-\left(\left(-\dfrac{21}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{45}\right)\right)\) et \( Y=\left(-\dfrac{17}{9}\right)\sqrt{20}+\left(\left(-\dfrac{37}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{43}{8}\right)\sqrt{45}\right)+5+\left(\dfrac{17}{4}\right)\sqrt{20}+\left(\dfrac{54}{7}\right)\sqrt{25}\) . 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{52}{5}\right)\sqrt{45}+\left(3\right)\sqrt{45}+\left(-9\right)\sqrt{20}\right)-\left(-4-\left(\left(-\dfrac{21}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{45}\right)\right)\right)+\left(\left(-\dfrac{17}{9}\right)\sqrt{20}+\left(\left(-\dfrac{37}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{43}{8}\right)\sqrt{45}\right)+5+\left(\dfrac{17}{4}\right)\sqrt{20}+\left(\dfrac{54}{7}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\dfrac{156}{5}\right)\sqrt{5}+\left(9\right)\sqrt{5}+\left(-18\right)\sqrt{5}\right)-\left(-4+\dfrac{105}{2}-\left(\left(-\dfrac{81}{2}\right)\sqrt{5}\right)\right)\right)+\left(\left(-\dfrac{34}{9}\right)\sqrt{5}-\dfrac{185}{9}-\dfrac{125}{3}-\left(\left(\dfrac{129}{8}\right)\sqrt{5}\right)+5+\left(\dfrac{17}{2}\right)\sqrt{5}+\dfrac{270}{7}\right)\\
&=&\left(\left(\dfrac{156}{5}\right)\sqrt{5}+\left(9\right)\sqrt{5}+\left(-18\right)\sqrt{5}\right)-\left(-4+\dfrac{105}{2}-\left(\left(-\dfrac{81}{2}\right)\sqrt{5}\right)\right)+\left(-\dfrac{34}{9}\right)\sqrt{5}-\dfrac{185}{9}-\dfrac{125}{3}-\left(\left(\dfrac{129}{8}\right)\sqrt{5}\right)+5+\left(\dfrac{17}{2}\right)\sqrt{5}+\dfrac{270}{7}\\
&=&\left(-\dfrac{10693}{360}\right)\sqrt{5}-\dfrac{8461}{126}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{52}{5}\right)\sqrt{45}+\left(3\right)\sqrt{45}+\left(-9\right)\sqrt{20}\right)-\left(-4-\left(\left(-\dfrac{21}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{45}\right)\right)\right)-\left(\left(-\dfrac{17}{9}\right)\sqrt{20}+\left(\left(-\dfrac{37}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{43}{8}\right)\sqrt{45}\right)+5+\left(\dfrac{17}{4}\right)\sqrt{20}+\left(\dfrac{54}{7}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\dfrac{156}{5}\right)\sqrt{5}+\left(9\right)\sqrt{5}+\left(-18\right)\sqrt{5}\right)-\left(-4+\dfrac{105}{2}-\left(\left(-\dfrac{81}{2}\right)\sqrt{5}\right)\right)\right)-\left(\left(-\dfrac{34}{9}\right)\sqrt{5}-\dfrac{185}{9}-\dfrac{125}{3}-\left(\left(\dfrac{129}{8}\right)\sqrt{5}\right)+5+\left(\dfrac{17}{2}\right)\sqrt{5}+\dfrac{270}{7}\right)\\
&=&\left(\left(-\dfrac{183}{10}\right)\sqrt{5}-\dfrac{97}{2}\right)-\left(\left(-\dfrac{821}{72}\right)\sqrt{5}-\dfrac{1175}{63}\right)\\
&=&\left(-\dfrac{183}{10}\right)\sqrt{5}-\dfrac{97}{2}+\left(\dfrac{821}{72}\right)\sqrt{5}+\dfrac{1175}{63}\\
&=&\left(-\dfrac{2483}{360}\right)\sqrt{5}-\dfrac{3761}{126}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{52}{5}\right)\sqrt{45}+\left(3\right)\sqrt{45}+\left(-9\right)\sqrt{20}\right)-\left(-4-\left(\left(-\dfrac{21}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{45}\right)\right)\right)\times\left(\left(-\dfrac{17}{9}\right)\sqrt{20}+\left(\left(-\dfrac{37}{9}\right)\sqrt{25}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{43}{8}\right)\sqrt{45}\right)+5+\left(\dfrac{17}{4}\right)\sqrt{20}+\left(\dfrac{54}{7}\right)\sqrt{25}\right)\\
&=&\left(\left(\left(\dfrac{156}{5}\right)\sqrt{5}+\left(9\right)\sqrt{5}+\left(-18\right)\sqrt{5}\right)-\left(-4+\dfrac{105}{2}-\left(\left(-\dfrac{81}{2}\right)\sqrt{5}\right)\right)\right)\times\left(\left(-\dfrac{34}{9}\right)\sqrt{5}-\dfrac{185}{9}-\dfrac{125}{3}-\left(\left(\dfrac{129}{8}\right)\sqrt{5}\right)+5+\left(\dfrac{17}{2}\right)\sqrt{5}+\dfrac{270}{7}\right)\\
&=&\left(\left(-\dfrac{183}{10}\right)\sqrt{5}-\dfrac{97}{2}\right)\left(\left(-\dfrac{821}{72}\right)\sqrt{5}-\dfrac{1175}{63}\right)\\
&=&\left(\dfrac{50081}{240}\right)\sqrt{25}+\left(\dfrac{901499}{1008}\right)\sqrt{5}+\dfrac{113975}{126}\\
&=&\dfrac{1963501}{1008}+\left(\dfrac{901499}{1008}\right)\sqrt{5}\\
\end{eqnarray*}