L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-8\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{67}{8}\right)\sqrt{25}\right)-0+3\right)\) et \( Y=\dfrac{81}{4}+\left(\dfrac{67}{7}\right)\sqrt{20}+\left(\dfrac{14}{3}\right)\sqrt{25}+\left(0\right)\sqrt{45}+\left(2\right)\sqrt{20}\) . 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(-8\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{67}{8}\right)\sqrt{25}\right)-0+3\right)\right)+\left(\dfrac{81}{4}+\left(\dfrac{67}{7}\right)\sqrt{20}+\left(\dfrac{14}{3}\right)\sqrt{25}+\left(0\right)\sqrt{45}+\left(2\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-24\right)\sqrt{5}\right)-\left(-\dfrac{335}{8}-0+3\right)\right)+\left(\dfrac{81}{4}+\left(\dfrac{134}{7}\right)\sqrt{5}+\dfrac{70}{3}+\left(0\right)\sqrt{5}+\left(4\right)\sqrt{5}\right)\\
&=&\left(\left(-24\right)\sqrt{5}\right)-\left(-\dfrac{335}{8}-0+3\right)+\dfrac{81}{4}+\left(\dfrac{134}{7}\right)\sqrt{5}+\dfrac{70}{3}+\left(0\right)\sqrt{5}+\left(4\right)\sqrt{5}\\
&=&\left(-\dfrac{6}{7}\right)\sqrt{5}+\dfrac{1979}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-8\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{67}{8}\right)\sqrt{25}\right)-0+3\right)\right)-\left(\dfrac{81}{4}+\left(\dfrac{67}{7}\right)\sqrt{20}+\left(\dfrac{14}{3}\right)\sqrt{25}+\left(0\right)\sqrt{45}+\left(2\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-24\right)\sqrt{5}\right)-\left(-\dfrac{335}{8}-0+3\right)\right)-\left(\dfrac{81}{4}+\left(\dfrac{134}{7}\right)\sqrt{5}+\dfrac{70}{3}+\left(0\right)\sqrt{5}+\left(4\right)\sqrt{5}\right)\\
&=&\left(\left(-24\right)\sqrt{5}+\dfrac{311}{8}\right)-\left(\dfrac{523}{12}+\left(\dfrac{162}{7}\right)\sqrt{5}\right)\\
&=&\left(-24\right)\sqrt{5}+\dfrac{311}{8}+-\dfrac{523}{12}+\left(-\dfrac{162}{7}\right)\sqrt{5}\\
&=&\left(-\dfrac{330}{7}\right)\sqrt{5}-\dfrac{113}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-8\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{67}{8}\right)\sqrt{25}\right)-0+3\right)\right)\times\left(\dfrac{81}{4}+\left(\dfrac{67}{7}\right)\sqrt{20}+\left(\dfrac{14}{3}\right)\sqrt{25}+\left(0\right)\sqrt{45}+\left(2\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-24\right)\sqrt{5}\right)-\left(-\dfrac{335}{8}-0+3\right)\right)\times\left(\dfrac{81}{4}+\left(\dfrac{134}{7}\right)\sqrt{5}+\dfrac{70}{3}+\left(0\right)\sqrt{5}+\left(4\right)\sqrt{5}\right)\\
&=&\left(\left(-24\right)\sqrt{5}+\dfrac{311}{8}\right)\left(\dfrac{523}{12}+\left(\dfrac{162}{7}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{4097}{28}\right)\sqrt{5}+\left(-\dfrac{3888}{7}\right)\sqrt{25}+\dfrac{162653}{96}\\
&=&\left(-\dfrac{4097}{28}\right)\sqrt{5}-\dfrac{727669}{672}\\
\end{eqnarray*}