L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(7\right)\sqrt{75}\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{12}\right)\right)-\left(\left(3\right)\sqrt{27}+\left(\dfrac{35}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{59}{7}\right)\sqrt{27}\right)\) et \( Y=\left(-\dfrac{6}{7}\right)\sqrt{27}\) . 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(7\right)\sqrt{75}\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{12}\right)\right)-\left(\left(3\right)\sqrt{27}+\left(\dfrac{35}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{59}{7}\right)\sqrt{27}\right)\right)+\left(\left(-\dfrac{6}{7}\right)\sqrt{27}\right)\\
&=&\left(\left(\left(\left(35\right)\sqrt{3}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)\right)-\left(\left(9\right)\sqrt{3}+\dfrac{105}{4}\right)-\left(\left(\dfrac{177}{7}\right)\sqrt{3}\right)\right)+\left(\left(-\dfrac{18}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(\left(35\right)\sqrt{3}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)\right)-\left(\left(9\right)\sqrt{3}+\dfrac{105}{4}\right)-\left(\left(\dfrac{177}{7}\right)\sqrt{3}\right)+\left(-\dfrac{18}{7}\right)\sqrt{3}\\
&=&\left(-\dfrac{327}{14}\right)\sqrt{3}-\dfrac{105}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(7\right)\sqrt{75}\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{12}\right)\right)-\left(\left(3\right)\sqrt{27}+\left(\dfrac{35}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{59}{7}\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{6}{7}\right)\sqrt{27}\right)\\
&=&\left(\left(\left(\left(35\right)\sqrt{3}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)\right)-\left(\left(9\right)\sqrt{3}+\dfrac{105}{4}\right)-\left(\left(\dfrac{177}{7}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{291}{14}\right)\sqrt{3}-\dfrac{105}{4}\right)-\left(\left(-\dfrac{18}{7}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{291}{14}\right)\sqrt{3}-\dfrac{105}{4}+\left(\dfrac{18}{7}\right)\sqrt{3}\\
&=&\left(-\dfrac{255}{14}\right)\sqrt{3}-\dfrac{105}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(7\right)\sqrt{75}\right)-\left(\left(\dfrac{13}{2}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{12}\right)\right)-\left(\left(3\right)\sqrt{27}+\left(\dfrac{35}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{59}{7}\right)\sqrt{27}\right)\right)\times\left(\left(-\dfrac{6}{7}\right)\sqrt{27}\right)\\
&=&\left(\left(\left(\left(35\right)\sqrt{3}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)\right)-\left(\left(9\right)\sqrt{3}+\dfrac{105}{4}\right)-\left(\left(\dfrac{177}{7}\right)\sqrt{3}\right)\right)\times\left(\left(-\dfrac{18}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{291}{14}\right)\sqrt{3}-\dfrac{105}{4}\right)\left(\left(-\dfrac{18}{7}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{2619}{49}\right)\sqrt{9}+\left(\dfrac{135}{2}\right)\sqrt{3}\\
&=&\dfrac{7857}{49}+\left(\dfrac{135}{2}\right)\sqrt{3}\\
\end{eqnarray*}