L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-2\right)\sqrt{45}+\left(\dfrac{14}{3}\right)\sqrt{25}-\dfrac{71}{8}+\dfrac{41}{2}-\left(\left(8\right)\sqrt{20}\right)+\left(8\right)\sqrt{20}+\dfrac{11}{4}+\left(-\dfrac{72}{7}\right)\sqrt{20}\) et \( Y=\left(\dfrac{15}{7}\right)\sqrt{125}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-2\right)\sqrt{45}+\left(\dfrac{14}{3}\right)\sqrt{25}-\dfrac{71}{8}+\dfrac{41}{2}-\left(\left(8\right)\sqrt{20}\right)+\left(8\right)\sqrt{20}+\dfrac{11}{4}+\left(-\dfrac{72}{7}\right)\sqrt{20}\right)+\left(\left(\dfrac{15}{7}\right)\sqrt{125}\right)\\
&=&\left(\left(-6\right)\sqrt{5}+\dfrac{70}{3}-\dfrac{71}{8}+\dfrac{41}{2}-\left(\left(16\right)\sqrt{5}\right)+\left(16\right)\sqrt{5}+\dfrac{11}{4}+\left(-\dfrac{144}{7}\right)\sqrt{5}\right)+\left(\left(\dfrac{75}{7}\right)\sqrt{5}\right)\\
&=&\left(-6\right)\sqrt{5}+\dfrac{70}{3}-\dfrac{71}{8}+\dfrac{41}{2}-\left(\left(16\right)\sqrt{5}\right)+\left(16\right)\sqrt{5}+\dfrac{11}{4}+\left(-\dfrac{144}{7}\right)\sqrt{5}+\left(\dfrac{75}{7}\right)\sqrt{5}\\
&=&\left(-\dfrac{111}{7}\right)\sqrt{5}+\dfrac{905}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-2\right)\sqrt{45}+\left(\dfrac{14}{3}\right)\sqrt{25}-\dfrac{71}{8}+\dfrac{41}{2}-\left(\left(8\right)\sqrt{20}\right)+\left(8\right)\sqrt{20}+\dfrac{11}{4}+\left(-\dfrac{72}{7}\right)\sqrt{20}\right)-\left(\left(\dfrac{15}{7}\right)\sqrt{125}\right)\\
&=&\left(\left(-6\right)\sqrt{5}+\dfrac{70}{3}-\dfrac{71}{8}+\dfrac{41}{2}-\left(\left(16\right)\sqrt{5}\right)+\left(16\right)\sqrt{5}+\dfrac{11}{4}+\left(-\dfrac{144}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{186}{7}\right)\sqrt{5}+\dfrac{905}{24}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{186}{7}\right)\sqrt{5}+\dfrac{905}{24}+\left(-\dfrac{75}{7}\right)\sqrt{5}\\
&=&\left(-\dfrac{261}{7}\right)\sqrt{5}+\dfrac{905}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-2\right)\sqrt{45}+\left(\dfrac{14}{3}\right)\sqrt{25}-\dfrac{71}{8}+\dfrac{41}{2}-\left(\left(8\right)\sqrt{20}\right)+\left(8\right)\sqrt{20}+\dfrac{11}{4}+\left(-\dfrac{72}{7}\right)\sqrt{20}\right)\times\left(\left(\dfrac{15}{7}\right)\sqrt{125}\right)\\
&=&\left(\left(-6\right)\sqrt{5}+\dfrac{70}{3}-\dfrac{71}{8}+\dfrac{41}{2}-\left(\left(16\right)\sqrt{5}\right)+\left(16\right)\sqrt{5}+\dfrac{11}{4}+\left(-\dfrac{144}{7}\right)\sqrt{5}\right)\times\left(\left(\dfrac{75}{7}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{186}{7}\right)\sqrt{5}+\dfrac{905}{24}\right)\left(\left(\dfrac{75}{7}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{13950}{49}\right)\sqrt{25}+\left(\dfrac{22625}{56}\right)\sqrt{5}\\
&=&-\dfrac{69750}{49}+\left(\dfrac{22625}{56}\right)\sqrt{5}\\
\end{eqnarray*}