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 \( X=\left(\dfrac{1}{5}\right)\sqrt{45}+\left(-9\right)\sqrt{45}+\left(-\dfrac{21}{4}\right)\sqrt{25}+\left(-5\right)\sqrt{45}+\left(0\right)\sqrt{20}+\left(5\right)\sqrt{20}+\left(-\dfrac{55}{8}\right)\sqrt{20}+\left(\left(-\dfrac{3}{4}\right)\sqrt{25}\right)-\left(\left(5\right)\sqrt{20}\right)+\left(\left(-7\right)\sqrt{25}\right)-\left(\left(8\right)\sqrt{25}\right)\) et \( Y=\left(\left(-\dfrac{15}{2}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{21}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{12}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{125}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{1}{5}\right)\sqrt{45}+\left(-9\right)\sqrt{45}+\left(-\dfrac{21}{4}\right)\sqrt{25}+\left(-5\right)\sqrt{45}+\left(0\right)\sqrt{20}+\left(5\right)\sqrt{20}+\left(-\dfrac{55}{8}\right)\sqrt{20}+\left(\left(-\dfrac{3}{4}\right)\sqrt{25}\right)-\left(\left(5\right)\sqrt{20}\right)+\left(\left(-7\right)\sqrt{25}\right)-\left(\left(8\right)\sqrt{25}\right)\right)+\left(\left(\left(-\dfrac{15}{2}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{21}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{12}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\dfrac{3}{5}\right)\sqrt{5}+\left(-27\right)\sqrt{5}-\dfrac{105}{4}+\left(-15\right)\sqrt{5}+\left(0\right)\sqrt{5}+\left(10\right)\sqrt{5}+\left(-\dfrac{55}{4}\right)\sqrt{5}-\dfrac{15}{4}-\left(\left(10\right)\sqrt{5}\right)-35-40\right)+\left(-\dfrac{75}{2}-\left(21-\left(\left(-\dfrac{36}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{4}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\dfrac{3}{5}\right)\sqrt{5}+\left(-27\right)\sqrt{5}-\dfrac{105}{4}+\left(-15\right)\sqrt{5}+\left(0\right)\sqrt{5}+\left(10\right)\sqrt{5}+\left(-\dfrac{55}{4}\right)\sqrt{5}-\dfrac{15}{4}-\left(\left(10\right)\sqrt{5}\right)-35-40-\dfrac{75}{2}-\left(21-\left(\left(-\dfrac{36}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{4}\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{4483}{70}\right)\sqrt{5}-\dfrac{327}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{1}{5}\right)\sqrt{45}+\left(-9\right)\sqrt{45}+\left(-\dfrac{21}{4}\right)\sqrt{25}+\left(-5\right)\sqrt{45}+\left(0\right)\sqrt{20}+\left(5\right)\sqrt{20}+\left(-\dfrac{55}{8}\right)\sqrt{20}+\left(\left(-\dfrac{3}{4}\right)\sqrt{25}\right)-\left(\left(5\right)\sqrt{20}\right)+\left(\left(-7\right)\sqrt{25}\right)-\left(\left(8\right)\sqrt{25}\right)\right)-\left(\left(\left(-\dfrac{15}{2}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{21}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{12}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\dfrac{3}{5}\right)\sqrt{5}+\left(-27\right)\sqrt{5}-\dfrac{105}{4}+\left(-15\right)\sqrt{5}+\left(0\right)\sqrt{5}+\left(10\right)\sqrt{5}+\left(-\dfrac{55}{4}\right)\sqrt{5}-\dfrac{15}{4}-\left(\left(10\right)\sqrt{5}\right)-35-40\right)-\left(-\dfrac{75}{2}-\left(21-\left(\left(-\dfrac{36}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{4}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(-\dfrac{1103}{20}\right)\sqrt{5}-105\right)-\left(-\dfrac{117}{2}+\left(-\dfrac{249}{28}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{1103}{20}\right)\sqrt{5}-105+\dfrac{117}{2}+\left(\dfrac{249}{28}\right)\sqrt{5}\\
&=&\left(-\dfrac{1619}{35}\right)\sqrt{5}-\dfrac{93}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{1}{5}\right)\sqrt{45}+\left(-9\right)\sqrt{45}+\left(-\dfrac{21}{4}\right)\sqrt{25}+\left(-5\right)\sqrt{45}+\left(0\right)\sqrt{20}+\left(5\right)\sqrt{20}+\left(-\dfrac{55}{8}\right)\sqrt{20}+\left(\left(-\dfrac{3}{4}\right)\sqrt{25}\right)-\left(\left(5\right)\sqrt{20}\right)+\left(\left(-7\right)\sqrt{25}\right)-\left(\left(8\right)\sqrt{25}\right)\right)\times\left(\left(\left(-\dfrac{15}{2}\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{21}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{12}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\dfrac{3}{5}\right)\sqrt{5}+\left(-27\right)\sqrt{5}-\dfrac{105}{4}+\left(-15\right)\sqrt{5}+\left(0\right)\sqrt{5}+\left(10\right)\sqrt{5}+\left(-\dfrac{55}{4}\right)\sqrt{5}-\dfrac{15}{4}-\left(\left(10\right)\sqrt{5}\right)-35-40\right)\times\left(-\dfrac{75}{2}-\left(21-\left(\left(-\dfrac{36}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{4}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(-\dfrac{1103}{20}\right)\sqrt{5}-105\right)\left(-\dfrac{117}{2}+\left(-\dfrac{249}{28}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{166401}{40}\right)\sqrt{5}+\left(\dfrac{274647}{560}\right)\sqrt{25}+\dfrac{12285}{2}\\
&=&\left(\dfrac{166401}{40}\right)\sqrt{5}+\dfrac{962607}{112}\\
\end{eqnarray*}