L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{36}{5}\right)\sqrt{125}+\left(\left(-\dfrac{1}{3}\right)\sqrt{45}\right)-\left(\left(4\right)\sqrt{25}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{45}\right)\) et \( Y=7-\left(\left(-\dfrac{75}{2}\right)\sqrt{20}+\left(-\dfrac{69}{8}\right)\sqrt{45}\right)-\left(\left(7\right)\sqrt{20}\right)\) . 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{36}{5}\right)\sqrt{125}+\left(\left(-\dfrac{1}{3}\right)\sqrt{45}\right)-\left(\left(4\right)\sqrt{25}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{45}\right)\right)+\left(7-\left(\left(-\dfrac{75}{2}\right)\sqrt{20}+\left(-\dfrac{69}{8}\right)\sqrt{45}\right)-\left(\left(7\right)\sqrt{20}\right)\right)\\
&=&\left(\left(-36\right)\sqrt{5}+\left(\left(-1\right)\sqrt{5}\right)-20-\left(\left(-1\right)\sqrt{5}\right)\right)+\left(7-\left(\left(-75\right)\sqrt{5}+\left(-\dfrac{207}{8}\right)\sqrt{5}\right)-\left(\left(14\right)\sqrt{5}\right)\right)\\
&=&\left(-36\right)\sqrt{5}+\left(\left(-1\right)\sqrt{5}\right)-20-\left(\left(-1\right)\sqrt{5}\right)+7-\left(\left(-75\right)\sqrt{5}+\left(-\dfrac{207}{8}\right)\sqrt{5}\right)-\left(\left(14\right)\sqrt{5}\right)\\
&=&\left(\dfrac{407}{8}\right)\sqrt{5}-13\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{36}{5}\right)\sqrt{125}+\left(\left(-\dfrac{1}{3}\right)\sqrt{45}\right)-\left(\left(4\right)\sqrt{25}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{45}\right)\right)-\left(7-\left(\left(-\dfrac{75}{2}\right)\sqrt{20}+\left(-\dfrac{69}{8}\right)\sqrt{45}\right)-\left(\left(7\right)\sqrt{20}\right)\right)\\
&=&\left(\left(-36\right)\sqrt{5}+\left(\left(-1\right)\sqrt{5}\right)-20-\left(\left(-1\right)\sqrt{5}\right)\right)-\left(7-\left(\left(-75\right)\sqrt{5}+\left(-\dfrac{207}{8}\right)\sqrt{5}\right)-\left(\left(14\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-36\right)\sqrt{5}-20\right)-\left(7+\left(\dfrac{695}{8}\right)\sqrt{5}\right)\\
&=&\left(-36\right)\sqrt{5}-20+-7+\left(-\dfrac{695}{8}\right)\sqrt{5}\\
&=&\left(-\dfrac{983}{8}\right)\sqrt{5}-27\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{36}{5}\right)\sqrt{125}+\left(\left(-\dfrac{1}{3}\right)\sqrt{45}\right)-\left(\left(4\right)\sqrt{25}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{45}\right)\right)\times\left(7-\left(\left(-\dfrac{75}{2}\right)\sqrt{20}+\left(-\dfrac{69}{8}\right)\sqrt{45}\right)-\left(\left(7\right)\sqrt{20}\right)\right)\\
&=&\left(\left(-36\right)\sqrt{5}+\left(\left(-1\right)\sqrt{5}\right)-20-\left(\left(-1\right)\sqrt{5}\right)\right)\times\left(7-\left(\left(-75\right)\sqrt{5}+\left(-\dfrac{207}{8}\right)\sqrt{5}\right)-\left(\left(14\right)\sqrt{5}\right)\right)\\
&=&\left(\left(-36\right)\sqrt{5}-20\right)\left(7+\left(\dfrac{695}{8}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{3979}{2}\right)\sqrt{5}+\left(-\dfrac{6255}{2}\right)\sqrt{25}-140\\
&=&\left(-\dfrac{3979}{2}\right)\sqrt{5}-\dfrac{31555}{2}\\
\end{eqnarray*}