L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{40}{3}\right)\sqrt{20}+\left(\dfrac{56}{3}\right)\sqrt{125}\right)-\dfrac{23}{4}-\left(0-\dfrac{73}{8}+\left(-\dfrac{60}{7}\right)\sqrt{45}+\left(-\dfrac{74}{7}\right)\sqrt{25}+\left(\dfrac{11}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{52}{3}\right)\sqrt{25}+\left(-\dfrac{5}{3}\right)\sqrt{25}+\dfrac{69}{5}\right)\) et \( Y=\left(6\right)\sqrt{20}\) . 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{20}+\left(\dfrac{56}{3}\right)\sqrt{125}\right)-\dfrac{23}{4}-\left(0-\dfrac{73}{8}+\left(-\dfrac{60}{7}\right)\sqrt{45}+\left(-\dfrac{74}{7}\right)\sqrt{25}+\left(\dfrac{11}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{52}{3}\right)\sqrt{25}+\left(-\dfrac{5}{3}\right)\sqrt{25}+\dfrac{69}{5}\right)\right)+\left(\left(6\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-\dfrac{80}{3}\right)\sqrt{5}+\left(\dfrac{280}{3}\right)\sqrt{5}\right)-\dfrac{23}{4}-\left(0-\dfrac{73}{8}+\left(-\dfrac{180}{7}\right)\sqrt{5}-\dfrac{370}{7}+\left(11\right)\sqrt{5}\right)-\left(-\dfrac{260}{3}-\dfrac{25}{3}+\dfrac{69}{5}\right)\right)+\left(\left(12\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{80}{3}\right)\sqrt{5}+\left(\dfrac{280}{3}\right)\sqrt{5}\right)-\dfrac{23}{4}-\left(0-\dfrac{73}{8}+\left(-\dfrac{180}{7}\right)\sqrt{5}-\dfrac{370}{7}+\left(11\right)\sqrt{5}\right)-\left(-\dfrac{260}{3}-\dfrac{25}{3}+\dfrac{69}{5}\right)+\left(12\right)\sqrt{5}\\
&=&\left(\dfrac{1961}{21}\right)\sqrt{5}+\dfrac{38481}{280}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{40}{3}\right)\sqrt{20}+\left(\dfrac{56}{3}\right)\sqrt{125}\right)-\dfrac{23}{4}-\left(0-\dfrac{73}{8}+\left(-\dfrac{60}{7}\right)\sqrt{45}+\left(-\dfrac{74}{7}\right)\sqrt{25}+\left(\dfrac{11}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{52}{3}\right)\sqrt{25}+\left(-\dfrac{5}{3}\right)\sqrt{25}+\dfrac{69}{5}\right)\right)-\left(\left(6\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-\dfrac{80}{3}\right)\sqrt{5}+\left(\dfrac{280}{3}\right)\sqrt{5}\right)-\dfrac{23}{4}-\left(0-\dfrac{73}{8}+\left(-\dfrac{180}{7}\right)\sqrt{5}-\dfrac{370}{7}+\left(11\right)\sqrt{5}\right)-\left(-\dfrac{260}{3}-\dfrac{25}{3}+\dfrac{69}{5}\right)\right)-\left(\left(12\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{1709}{21}\right)\sqrt{5}+\dfrac{38481}{280}\right)-\left(\left(12\right)\sqrt{5}\right)\\
&=&\left(\dfrac{1709}{21}\right)\sqrt{5}+\dfrac{38481}{280}+\left(-12\right)\sqrt{5}\\
&=&\left(\dfrac{1457}{21}\right)\sqrt{5}+\dfrac{38481}{280}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{40}{3}\right)\sqrt{20}+\left(\dfrac{56}{3}\right)\sqrt{125}\right)-\dfrac{23}{4}-\left(0-\dfrac{73}{8}+\left(-\dfrac{60}{7}\right)\sqrt{45}+\left(-\dfrac{74}{7}\right)\sqrt{25}+\left(\dfrac{11}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{52}{3}\right)\sqrt{25}+\left(-\dfrac{5}{3}\right)\sqrt{25}+\dfrac{69}{5}\right)\right)\times\left(\left(6\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-\dfrac{80}{3}\right)\sqrt{5}+\left(\dfrac{280}{3}\right)\sqrt{5}\right)-\dfrac{23}{4}-\left(0-\dfrac{73}{8}+\left(-\dfrac{180}{7}\right)\sqrt{5}-\dfrac{370}{7}+\left(11\right)\sqrt{5}\right)-\left(-\dfrac{260}{3}-\dfrac{25}{3}+\dfrac{69}{5}\right)\right)\times\left(\left(12\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{1709}{21}\right)\sqrt{5}+\dfrac{38481}{280}\right)\left(\left(12\right)\sqrt{5}\right)\\
&=&\left(\dfrac{6836}{7}\right)\sqrt{25}+\left(\dfrac{115443}{70}\right)\sqrt{5}\\
&=&\dfrac{34180}{7}+\left(\dfrac{115443}{70}\right)\sqrt{5}\\
\end{eqnarray*}