L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)-\left(\left(-2\right)\sqrt{9}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{77}{3}\right)\sqrt{9}\right)\) et \( Y=\left(\left(\dfrac{21}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{27}\right)+\left(\left(0\right)\sqrt{75}\right)-\left(\left(5\right)\sqrt{12}\right)-2\) . 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(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)-\left(\left(-2\right)\sqrt{9}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{77}{3}\right)\sqrt{9}\right)\right)+\left(\left(\left(\dfrac{21}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{27}\right)+\left(\left(0\right)\sqrt{75}\right)-\left(\left(5\right)\sqrt{12}\right)-2\right)\\
&=&\left(\left(\left(\left(-43\right)\sqrt{3}\right)+6-\dfrac{99}{4}\right)-\dfrac{27}{2}-77\right)+\left(\left(\left(\dfrac{105}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{3}\right)\sqrt{3}\right)+\left(\left(0\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)-2\right)\\
&=&\left(\left(\left(-43\right)\sqrt{3}\right)+6-\dfrac{99}{4}\right)-\dfrac{27}{2}-77+\left(\left(\dfrac{105}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{3}\right)\sqrt{3}\right)+\left(\left(0\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)-2\\
&=&\left(-\dfrac{437}{24}\right)\sqrt{3}-\dfrac{445}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)-\left(\left(-2\right)\sqrt{9}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{77}{3}\right)\sqrt{9}\right)\right)-\left(\left(\left(\dfrac{21}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{27}\right)+\left(\left(0\right)\sqrt{75}\right)-\left(\left(5\right)\sqrt{12}\right)-2\right)\\
&=&\left(\left(\left(\left(-43\right)\sqrt{3}\right)+6-\dfrac{99}{4}\right)-\dfrac{27}{2}-77\right)-\left(\left(\left(\dfrac{105}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{3}\right)\sqrt{3}\right)+\left(\left(0\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)-2\right)\\
&=&\left(\left(-43\right)\sqrt{3}-\dfrac{437}{4}\right)-\left(\left(\dfrac{595}{24}\right)\sqrt{3}-2\right)\\
&=&\left(-43\right)\sqrt{3}-\dfrac{437}{4}+\left(-\dfrac{595}{24}\right)\sqrt{3}+2\\
&=&\left(-\dfrac{1627}{24}\right)\sqrt{3}-\dfrac{429}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{43}{5}\right)\sqrt{75}\right)-\left(\left(-2\right)\sqrt{9}\right)-\left(\left(\dfrac{33}{4}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{77}{3}\right)\sqrt{9}\right)\right)\times\left(\left(\left(\dfrac{21}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{27}\right)+\left(\left(0\right)\sqrt{75}\right)-\left(\left(5\right)\sqrt{12}\right)-2\right)\\
&=&\left(\left(\left(\left(-43\right)\sqrt{3}\right)+6-\dfrac{99}{4}\right)-\dfrac{27}{2}-77\right)\times\left(\left(\left(\dfrac{105}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{65}{3}\right)\sqrt{3}\right)+\left(\left(0\right)\sqrt{3}\right)-\left(\left(10\right)\sqrt{3}\right)-2\right)\\
&=&\left(\left(-43\right)\sqrt{3}-\dfrac{437}{4}\right)\left(\left(\dfrac{595}{24}\right)\sqrt{3}-2\right)\\
&=&\left(-\dfrac{25585}{24}\right)\sqrt{9}+\left(-\dfrac{251759}{96}\right)\sqrt{3}+\dfrac{437}{2}\\
&=&-\dfrac{23837}{8}+\left(-\dfrac{251759}{96}\right)\sqrt{3}\\
\end{eqnarray*}