L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{37}{7}\right)\sqrt{75}\) et \( Y=\left(\left(-\dfrac{52}{3}\right)\sqrt{75}\right)-\left(\left(\left(-6\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{75}\right)\right)-\left(\left(8\right)\sqrt{12}+\left(-\dfrac{29}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{27}+\left(8\right)\sqrt{75}+\left(\dfrac{19}{8}\right)\sqrt{27}+1\right)\) . 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{37}{7}\right)\sqrt{75}\right)+\left(\left(\left(-\dfrac{52}{3}\right)\sqrt{75}\right)-\left(\left(\left(-6\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{75}\right)\right)-\left(\left(8\right)\sqrt{12}+\left(-\dfrac{29}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{27}+\left(8\right)\sqrt{75}+\left(\dfrac{19}{8}\right)\sqrt{27}+1\right)\right)\\
&=&\left(\left(-\dfrac{185}{7}\right)\sqrt{3}\right)+\left(\left(\left(-\dfrac{260}{3}\right)\sqrt{3}\right)-\left(\left(\left(-12\right)\sqrt{3}\right)-\left(\left(-35\right)\sqrt{3}\right)\right)-\left(\left(16\right)\sqrt{3}+\left(-\dfrac{145}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{3}+\left(40\right)\sqrt{3}+\left(\dfrac{57}{8}\right)\sqrt{3}+1\right)\right)\\
&=&\left(-\dfrac{185}{7}\right)\sqrt{3}+\left(\left(-\dfrac{260}{3}\right)\sqrt{3}\right)-\left(\left(\left(-12\right)\sqrt{3}\right)-\left(\left(-35\right)\sqrt{3}\right)\right)-\left(\left(16\right)\sqrt{3}+\left(-\dfrac{145}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{3}+\left(40\right)\sqrt{3}+\left(\dfrac{57}{8}\right)\sqrt{3}+1\right)\\
&=&\left(-\dfrac{26119}{168}\right)\sqrt{3}-1\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{37}{7}\right)\sqrt{75}\right)-\left(\left(\left(-\dfrac{52}{3}\right)\sqrt{75}\right)-\left(\left(\left(-6\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{75}\right)\right)-\left(\left(8\right)\sqrt{12}+\left(-\dfrac{29}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{27}+\left(8\right)\sqrt{75}+\left(\dfrac{19}{8}\right)\sqrt{27}+1\right)\right)\\
&=&\left(\left(-\dfrac{185}{7}\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{260}{3}\right)\sqrt{3}\right)-\left(\left(\left(-12\right)\sqrt{3}\right)-\left(\left(-35\right)\sqrt{3}\right)\right)-\left(\left(16\right)\sqrt{3}+\left(-\dfrac{145}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{3}+\left(40\right)\sqrt{3}+\left(\dfrac{57}{8}\right)\sqrt{3}+1\right)\right)\\
&=&\left(\left(-\dfrac{185}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{3097}{24}\right)\sqrt{3}-1\right)\\
&=&\left(-\dfrac{185}{7}\right)\sqrt{3}+\left(\dfrac{3097}{24}\right)\sqrt{3}+1\\
&=&\left(\dfrac{17239}{168}\right)\sqrt{3}+1\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{37}{7}\right)\sqrt{75}\right)\times\left(\left(\left(-\dfrac{52}{3}\right)\sqrt{75}\right)-\left(\left(\left(-6\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{75}\right)\right)-\left(\left(8\right)\sqrt{12}+\left(-\dfrac{29}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{27}+\left(8\right)\sqrt{75}+\left(\dfrac{19}{8}\right)\sqrt{27}+1\right)\right)\\
&=&\left(\left(-\dfrac{185}{7}\right)\sqrt{3}\right)\times\left(\left(\left(-\dfrac{260}{3}\right)\sqrt{3}\right)-\left(\left(\left(-12\right)\sqrt{3}\right)-\left(\left(-35\right)\sqrt{3}\right)\right)-\left(\left(16\right)\sqrt{3}+\left(-\dfrac{145}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{3}+\left(40\right)\sqrt{3}+\left(\dfrac{57}{8}\right)\sqrt{3}+1\right)\right)\\
&=&\left(\left(-\dfrac{185}{7}\right)\sqrt{3}\right)\left(\left(-\dfrac{3097}{24}\right)\sqrt{3}-1\right)\\
&=&\left(\dfrac{572945}{168}\right)\sqrt{9}+\left(\dfrac{185}{7}\right)\sqrt{3}\\
&=&\dfrac{572945}{56}+\left(\dfrac{185}{7}\right)\sqrt{3}\\
\end{eqnarray*}