L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(7\right)\sqrt{75}\right)-\left(-\dfrac{43}{7}+\left(\dfrac{61}{6}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{67}{6}\right)\sqrt{75}\right)+\dfrac{29}{4}-\dfrac{5}{6}-\left(\left(\dfrac{69}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{12}\right)\right)\) et \( Y=\left(-\dfrac{32}{5}\right)\sqrt{75}\) . 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(7\right)\sqrt{75}\right)-\left(-\dfrac{43}{7}+\left(\dfrac{61}{6}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{67}{6}\right)\sqrt{75}\right)+\dfrac{29}{4}-\dfrac{5}{6}-\left(\left(\dfrac{69}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{12}\right)\right)\right)+\left(\left(-\dfrac{32}{5}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(35\right)\sqrt{3}\right)-\left(-\dfrac{43}{7}+\left(\dfrac{61}{3}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{335}{6}\right)\sqrt{3}\right)+\dfrac{29}{4}-\dfrac{5}{6}-\left(\left(\dfrac{138}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{70}{9}\right)\sqrt{3}\right)\right)\right)+\left(\left(-32\right)\sqrt{3}\right)\\
&=&\left(\left(35\right)\sqrt{3}\right)-\left(-\dfrac{43}{7}+\left(\dfrac{61}{3}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{335}{6}\right)\sqrt{3}\right)+\dfrac{29}{4}-\dfrac{5}{6}-\left(\left(\dfrac{138}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{70}{9}\right)\sqrt{3}\right)\right)+\left(-32\right)\sqrt{3}\\
&=&\left(-\dfrac{5755}{126}\right)\sqrt{3}-\dfrac{23}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(7\right)\sqrt{75}\right)-\left(-\dfrac{43}{7}+\left(\dfrac{61}{6}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{67}{6}\right)\sqrt{75}\right)+\dfrac{29}{4}-\dfrac{5}{6}-\left(\left(\dfrac{69}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{12}\right)\right)\right)-\left(\left(-\dfrac{32}{5}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(35\right)\sqrt{3}\right)-\left(-\dfrac{43}{7}+\left(\dfrac{61}{3}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{335}{6}\right)\sqrt{3}\right)+\dfrac{29}{4}-\dfrac{5}{6}-\left(\left(\dfrac{138}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{70}{9}\right)\sqrt{3}\right)\right)\right)-\left(\left(-32\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{1723}{126}\right)\sqrt{3}-\dfrac{23}{84}\right)-\left(\left(-32\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{1723}{126}\right)\sqrt{3}-\dfrac{23}{84}+\left(32\right)\sqrt{3}\\
&=&\left(\dfrac{2309}{126}\right)\sqrt{3}-\dfrac{23}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(7\right)\sqrt{75}\right)-\left(-\dfrac{43}{7}+\left(\dfrac{61}{6}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{67}{6}\right)\sqrt{75}\right)+\dfrac{29}{4}-\dfrac{5}{6}-\left(\left(\dfrac{69}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{12}\right)\right)\right)\times\left(\left(-\dfrac{32}{5}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(35\right)\sqrt{3}\right)-\left(-\dfrac{43}{7}+\left(\dfrac{61}{3}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{335}{6}\right)\sqrt{3}\right)+\dfrac{29}{4}-\dfrac{5}{6}-\left(\left(\dfrac{138}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{70}{9}\right)\sqrt{3}\right)\right)\right)\times\left(\left(-32\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{1723}{126}\right)\sqrt{3}-\dfrac{23}{84}\right)\left(\left(-32\right)\sqrt{3}\right)\\
&=&\left(\dfrac{27568}{63}\right)\sqrt{9}+\left(\dfrac{184}{21}\right)\sqrt{3}\\
&=&\dfrac{27568}{21}+\left(\dfrac{184}{21}\right)\sqrt{3}\\
\end{eqnarray*}